Finite point configurations in Euclidean space I: an overview

In the past few years, my co-authors and I have been trying to answer the following basic question: How large does the Hausdorff dimension of a subset of need to be to ensure that it contains “most” finite point configurations of a given type?

A classical example of this question is the Falconer distance problem. It asks how large the Hausdorff dimension of , need to be to ensure that the Lebesgue measure of the distance set is positive. The conjectured dimensional threshold is . The best known threshold, due to Wolff in two dimensions and to Erdogan in higher dimensions is .

As we mention in several of the previous posts, the discrete analog of this problem is the Erdos distance problem which asks for the number of distances determined by a finite point set in , consisting of points. The conjecture is , up to a logarithmic factor in two dimensions. The two dimensional conjecture was recently proved in a brilliant paper by Larry Guth and Nets Katz. In higher dimensions, the best known result, due to Solymosi and Vu is, roughly, .

The possibility of a direct connection between the Erdos and Falconer problems is a fascinating subject that I have commented on in previous posts. It will come up again later in this post and its sequels as the question is still in many ways unresolved.

The distance set problem, whether discrete or continuous, can be viewed as the question about two point configurations. After all, a two point configuration is congruent to a two point configuration if and only if , where denotes the standard Euclidean distance. This begs the question of what happens with finite point configurations involving more points. In the discrete context, this questions has been explored in a series of papers by Erdos and Purdy and is exposed very nicely in a book, entitled “Research Problems in Discrete Geometry”, by Brass, Moser and Pach.

Let us fix some definitions. We say that two -simplexes and are congruent if there exists and such that . The reason for the convention where a -simplex contains points is that a two point configuration, for example, is a triangle, which is a two-dimensional object.

In discrete geometry, one of the basic questions about finite point configurations is the following. Let denote the maximum number of congruent copies of a given point configuration among points in . When and , this is the classical Erdos single distance problem, which asks how many times a given distance may arise among points in . Erdos conjectured that . The best result to date, which follows from a variant of the Szemeredi-Trotter incidence theorem, is . No progress on this problem, in either the positive or negative direction, has been achieved, to the best of my knowledge, since the early 80s.

The problem of estimating is quite frustrating. For a given triangle to repeat, each side-lengths has to repeat, so, in particular, , which we know to be bounded by . If the Erdos single distance conjecture is true, then , which would be essentially best possible. However, if the Erdos single distance conjecture is not true, there is a possibility that is much smaller than . Since for a triangle to repeat, all three sides need to repeat, one might think that it should not bee to hard to improve the estimate. Unfortunately, there has been no progress on this question for general point sets to the best of my knowledge. For point sets satisfying some additional structural assumptions, such as homogeneous point sets studied by Laba, Solymosi, Vu and others. there has been some progress via analytic estimates, and this is where our narrative now takes us.

In one of the previous posts, I described my recent result with Allan Greenleaf, entitled “On three point configurations determined by subsets of the Euclidean plane, the associated bilinear operator and applications to discrete geometry” (http://arxiv.org/pdf/1009.2471.pdf), where we prove that if is a Frostman measure on a compact set of Hausdorff dimension , then

for any non-zero satisfying the triangle inequality. From this one can easily deduce that if the Hausdorff dimension of is greater than , then the three dimensional Lebesgue measure of , the set of non-congruent triangles determined by is positive.

Using a variant of a conversion mechanism I developed with S. Hofmann and I. Laba a few years ago, one can use this result to improve the upper bound for restricted to finite subsets of the plane satisfying certain additional structural assumptions. The result applies to homogeneous sets, but the range of usefulness is actually much wider. Here is the basic idea, introduced by Iosevich, Rudnev and Uriarte-Tuero in “Theory of dimension for large discrete sets and applications” (http://arxiv.org/pdf/0707.1322.pdf). Let be a set of points in the unit square in the plane. Thicken each point by , , and let denote the obvious probability measure on the resulting union of disks. Assume that the disks are disjoint. We say that is -adaptable, if

, which is the quantitative way of capturing the idea that the Hausdorff dimension of the set of disks above is uniformly in .

If we consider -adaptable sets with , we can improve that bound for in this context to

.

In the cases when , almost nothing is known in the discrete context about . This is one of the motivations for my paper in preparation with Loukas Grafakos, Allan Greenleaf and Eyvindur Palsson on multi-linear generalized Radon transforms and applications to geometric measure theory/combinatorics. In this paper, to be described in the next installment of this post, we prove certain multi-linear inequalities and use them to prove inequalities of the type

, where is a Frostman measure on a subset of of a Hausdorff dimension exceeding

.

We then use this inequality to contain non-trivial upper bounds for in the context of -adaptable point sets. This and much more is subject of “Finite point configurations in Euclidean space II: multi-linear generalized Radon transforms”, to be posted in the coming days.

We should also note that much work has been done on this problem in the vector spaces over finite fields, requiring some very different tools. For example, Michael Bennett, Jonathan Pakianathan and I recently obtained a non-trivial exponent for the distribution of triangles in two-dimensional vector spaces over finite fields by following the Elekes-Sharir paradigm used by Guth and Katz to resolve the Erdos distance conjecture in the plane. This will be a subject of a separate post in this series.

Some links to related papers not explicitly mentioned above:

On the Mattila-Sjolin theorem for distance sets by Alex Iosevich, Mihalis Mourgoglou, Krystal Taylor (http://arxiv.org/pdf/1110.6805.pdf)

–

On angles determined by fractal subsets of the Euclidean space via Sobolev bounds for bi-linear operators by Alex Iosevich, Mihalis Mourgoglou, Eyvindur Palsson (http://arxiv.org/pdf/1110.6792.pdf)

–

On volumes determined by subsets of Euclidean space by Allan Greenleaf, Alex Iosevich, Mihalis Mourgoglou (http://arxiv.org/pdf/1110.6790.pdf)

–

Multi-parameter projection theorems with applications to sums-products and finite point configurations in the Euclidean setting by B. Erdoğan, D. Hart, A. Iosevich (http://arxiv.org/pdf/1106.5544.pdf)

–

On sets of directions determined by subsets of ${\Bbb R}^d$ by Alex Iosevich, Mihalis Mourgoglou, Steven Senger (http://arxiv.org/pdf/1009.4169.pdf)

–

Sharpness of Falconer’s estimate in continuous and arithmetic settings, geometric incidence theorems and distribution of lattice points in convex domains Alex Iosevich, Steven Senger (http://arxiv.org/pdf/1006.1397.pdf)

10 reasons the U.S. is no longer the land of the free By Jonathan Turley, Published: January 13

Every year, the State Department issues reports on individual rights in other countries, monitoring the passage of restrictive laws and regulations around the world. Iran, for example, has been criticized for denying fair public trials and limiting privacy, while Russia has been taken to task for undermining due process. Other countries have been condemned for the use of secret evidence and torture.

Even as we pass judgment on countries we consider unfree, Americans remain confident that any definition of a free nation must include their own — the land of free. Yet, the laws and practices of the land should shake that confidence. In the decade since Sept. 11, 2001, this country has comprehensively reduced civil liberties in the name of an expanded security state. The most recent example of this was the National Defense Authorization Act, signed Dec. 31, which allows for the indefinite detention of citizens. At what point does the reduction of individual rights in our country change how we define ourselves?

While each new national security power Washington has embraced was controversial when enacted, they are often discussed in isolation. But they don’t operate in isolation. They form a mosaic of powers under which our country could be considered, at least in part, authoritarian. Americans often proclaim our nation as a symbol of freedom to the world while dismissing nations such as Cuba and China as categorically unfree. Yet, objectively, we may be only half right. Those countries do lack basic individual rights such as due process, placing them outside any reasonable definition of “free,” but the United States now has much more in common with such regimes than anyone may like to admit.

These countries also have constitutions that purport to guarantee freedoms and rights. But their governments have broad discretion in denying those rights and few real avenues for challenges by citizens — precisely the problem with the new laws in this country.

The list of powers acquired by the U.S. government since 9/11 puts us in rather troubling company.

Assassination of U.S. citizens

President Obama has claimed, as President George W. Bush did before him, the right to order the killing of any citizen considered a terrorist or an abettor of terrorism. Last year, he approved the killing of U.S. citizen Anwar al-Awlaqi and another citizen under this claimed inherent authority. Last month, administration officials affirmed that power, stating that the president can order the assassination of any citizen whom he considers allied with terrorists. (Nations such as Nigeria, Iran and Syria have been routinely criticized for extrajudicial killings of enemies of the state.)

Indefinite detention

Under the law signed last month, terrorism suspects are to be held by the military; the president also has the authority to indefinitely detain citizens accused of terrorism. While the administration claims that this provision only codified existing law, experts widely contest this view, and the administration has opposed efforts to challenge such authority in federal courts. The government continues to claim the right to strip citizens of legal protections based on its sole discretion. (China recently codified a more limited detention law for its citizens, while countries such as Cambodia have been singled out by the United States for “prolonged detention.”)

Arbitrary justice

The president now decides whether a person will receive a trial in the federal courts or in a military tribunal, a system that has been ridiculed around the world for lacking basic due process protections. Bush claimed this authority in 2001, and Obama has continued the practice. (Egypt and China have been denounced for maintaining separate military justice systems for selected defendants, including civilians.)

Warrantless searches

The president may now order warrantless surveillance, including a new capability to force companies and organizations to turn over information on citizens’ finances, communications and associations. Bush acquired this sweeping power under the Patriot Act in 2001, and in 2011, Obama extended the power, including searches of everything from business documents to library records. The government can use “national security letters” to demand, without probable cause, that organizations turn over information on citizens — and order them not to reveal the disclosure to the affected party. (Saudi Arabia and Pakistan operate under laws that allow the government to engage in widespread discretionary surveillance.)

Secret evidence

The government now routinely uses secret evidence to detain individuals and employs secret evidence in federal and military courts. It also forces the dismissal of cases against the United States by simply filing declarations that the cases would make the government reveal classified information that would harm national security — a claim made in a variety of privacy lawsuits and largely accepted by federal judges without question. Even legal opinions, cited as the basis for the government’s actions under the Bush and Obama administrations, have been classified. This allows the government to claim secret legal arguments to support secret proceedings using secret evidence. In addition, some cases never make it to court at all. The federal courts routinely deny constitutional challenges to policies and programs under a narrow definition of standing to bring a case.

War crimes

The world clamored for prosecutions of those responsible for waterboarding terrorism suspects during the Bush administration, but the Obama administration said in 2009 that it would not allow CIA employees to be investigated or prosecuted for such actions. This gutted not just treaty obligations but the Nuremberg principles of international law. When courts in countries such as Spain moved to investigate Bush officials for war crimes, the Obama administration reportedly urged foreign officials not to allow such cases to proceed, despite the fact that the United States has long claimed the same authority with regard to alleged war criminals in other countries. (Various nations have resisted investigations of officials accused of war crimes and torture. Some, such as Serbia and Chile, eventually relented to comply with international law; countries that have denied independent investigations include Iran, Syria and China.)

Secret court

The government has increased its use of the secret Foreign Intelligence Surveillance Court, which has expanded its secret warrants to include individuals deemed to be aiding or abetting hostile foreign governments or organizations. In 2011, Obama renewed these powers, including allowing secret searches of individuals who are not part of an identifiable terrorist group. The administration has asserted the right to ignore congressional limits on such surveillance. (Pakistan places national security surveillance under the unchecked powers of the military or intelligence services.)

Immunity from judicial review

Like the Bush administration, the Obama administration has successfully pushed for immunity for companies that assist in warrantless surveillance of citizens, blocking the ability of citizens to challenge the violation of privacy. (Similarly, China has maintained sweeping immunity claims both inside and outside the country and routinely blocks lawsuits against private companies.)

Continual monitoring of citizens

The Obama administration has successfully defended its claim that it can use GPS devices to monitor every move of targeted citizens without securing any court order or review. (Saudi Arabia has installed massive public surveillance systems, while Cuba is notorious for active monitoring of selected citizens.)

Extraordinary renditions

The government now has the ability to transfer both citizens and noncitizens to another country under a system known as extraordinary rendition, which has been denounced as using other countries, such as Syria, Saudi Arabia, Egypt and Pakistan, to torture suspects. The Obama administration says it is not continuing the abuses of this practice under Bush, but it insists on the unfettered right to order such transfers — including the possible transfer of U.S. citizens.

These new laws have come with an infusion of money into an expanded security system on the state and federal levels, including more public surveillance cameras, tens of thousands of security personnel and a massive expansion of a terrorist-chasing bureaucracy.

Some politicians shrug and say these increased powers are merely a response to the times we live in. Thus, Sen. Lindsey Graham (R-S.C.) could declare in an interview last spring without objection that “free speech is a great idea, but we’re in a war.” Of course, terrorism will never “surrender” and end this particular “war.”

Other politicians rationalize that, while such powers may exist, it really comes down to how they are used. This is a common response by liberals who cannot bring themselves to denounce Obama as they did Bush. Sen. Carl Levin (D-Mich.), for instance, has insisted that Congress is not making any decision on indefinite detention: “That is a decision which we leave where it belongs — in the executive branch.”

And in a signing statement with the defense authorization bill, Obama said he does not intend to use the latest power to indefinitely imprison citizens. Yet, he still accepted the power as a sort of regretful autocrat.

An authoritarian nation is defined not just by the use of authoritarian powers, but by the ability to use them. If a president can take away your freedom or your life on his own authority, all rights become little more than a discretionary grant subject to executive will.

The framers lived under autocratic rule and understood this danger better than we do. James Madison famously warned that we needed a system that did not depend on the good intentions or motivations of our rulers: “If men were angels, no government would be necessary.”

Benjamin Franklin was more direct. In 1787, a Mrs. Powel confronted Franklin after the signing of the Constitution and asked, “Well, Doctor, what have we got — a republic or a monarchy?” His response was a bit chilling: “A republic, Madam, if you can keep it.”

Since 9/11, we have created the very government the framers feared: a government with sweeping and largely unchecked powers resting on the hope that they will be used wisely.

The indefinite-detention provision in the defense authorization bill seemed to many civil libertarians like a betrayal by Obama. While the president had promised to veto the law over that provision, Levin, a sponsor of the bill, disclosed on the Senate floor that it was in fact the White House that approved the removal of any exception for citizens from indefinite detention.

Dishonesty from politicians is nothing new for Americans. The real question is whether we are lying to ourselves when we call this country the land of the free.
Jonathan Turley is the Shapiro professor of public interest law at George Washington University.

Full Interview with Holocaust historian Deborah Lipstadt

Full Interview with Holocaust historian Deborah Lipstadt
Talking with Haaretz, prominent scholar discusses effect of Eichmann trial of perception of the Holocaust, as well as recent comments by U.S. envoy to Belgium Howard Gutman.

By Chemi Shalev

Q. 50 years later, what was the significance of the Eichmann trial, in your eyes?
Lipstadt: The Eichmann trial was a pivotal moment in the history of Israel, in the history of Zionism. One of the most striking things about it, for an Israeli audience – over and above his capture and over and above the trial itself – is the beginning of a change in the Israelis’ attitude towards Shoah survivors. I remember when I came to Israel in 1966 I still heard the word “sabonim” (soap) for survivors – today that’s inconceivable.

If you go online and you look at the tapes of the trial, you see young men and women who stood there and gave testimony bekavod (with honor). And it was clear that these were not “ghetto Jews”, they were not failures or somehow fatally flawed; these were people who were chronologically and geographically challenged – they were at the wrong place at the wrong time. And the irony of course is that in those years, Israelis forgot that these same people who were denigrated and called sabonim, when they got off the boats – they were told “Hello, now you’re an Israeli, here’s a rifle, go and fight in Latroun.”

But Haim Guri and Moshe Shamir and the editorial board of Davar – they all write about how shocked they were by what they had heard at the trial. Now, it’s not true that there hadn’t been any talk of the Holocaust – there had been a lot of talk in the fifties, there was [Israel] Kastner [the Hungarian Jewish leader who was accused of collaborating with the Nazis]- his trial, his appeal and his murder; there was a big Knesset debate and a public battle over the statute of Yad Vashem – the professors wanted it to be a research institute and the survivors wanted it to be a memorial; there was the 1950 law on the collaborators, the debates over reparations from, and the relations with, Germany. So why were they amazed? What is it that surprises them?

I think what happened at the trial was that it personalized the Shoah. If you sat there day after day, you heard personal stories that you hadn’t heard before. And I think that began to change things. I believe it’s Leora Bilsky of Tel Aviv University who said that in the wake of the trial there was more tolerance in Israel – maybe it’s not there anymore today, I don’t know.

Israel was a little less –maybe for the moment because I think it changes in 1967 – “כוחי ועוצם ידי עשה לי את החיל הזה” (“My power and the might of my hand have gotten me this wealth”, Deuteronomy 8, 17). I’ve always liked that because it is said in the Torah as a negative, but people turn it around. I think that Gideon Hausner, after talking to survivors, understood that this “why didn’t they fight back kind of thing”, wasn’t legitimate.

There’s a powerful anecdote from the trial about Gabriel Bach, who later served on Israel’s Supreme Court, who was Hausner’s assistant but also the legal adviser of Bureau 06 that questioned Eichmann. He had a lot of contact with Eichmann, he saw him every day when they were questioning him and he saw him at the trial. Bach said that for him, the most powerful moment at the trial was not Eichmann’s testimony and not the cross-examination – but the first day of the trial when the bailiff shouts “Beit Hamishpat” and the three judges walk in and Eichmann stands at attention before three judges of Israel, with the symbol of Israel behind them. If you look at the clips that Yad Vashem has put on YouTube, when the judges walk in, and everybody is looking at the judges – Bach is looking at Eichmann.

Q. That was one of the positive things you said about Hanna Arendt, that she was the one who remarked that “for the first time since the year 70, when Jerusalem was destroyed by the Romans, Jews were able to sit in judgment on crimes committed against their own people.”

Lipstadt: I read that, especially to audiences that are poised to hate her and I ask “do you know where I read that first”? I belong to an Orthodox synagogue in Atlanta, it’s very low key, and when I’m there on Tisha B’Av they ask me to give a talk in the afternoon, on the Shoah, because Rabbi Soloveichik said that this is the proper day to talk about the Shoah, which was also [Menachem] Begin’s view. So it’s usually 7 o’clock, because it’s August and it’s a long day, and these are people who on some level are truly mourning the destruction of the temple, they really feel it, and I gave a talk on Eichmann and the following year I gave a talk on Arendt, and I say: I’m going to give you this quote, and it’s really significant because you’re all mourning the loss of the Temple in 70A.D., and I ask – now guess who said it? Someone says Hausner and someone says Ben Gurion and I say – well, it’s Arendt. She was cruel, she was mean, but she also captured something.

And you know how I know I got Arendt right in my book? Because the people who love Arendt hate the chapter on her, and the people who hate Arendt think I was soft on her.

Q. I read in one of the reviews that you “damn her with praise”, but in the end you destroy her. You may be nicer about it…but the result is the same.

Lipstadt: Well, you know who I hate more than Arendt? There’s the [1999] movie “The Specialist” a French documentary by a former Israeli, Eyal Sivan, and it takes Arendt farther than she would have ever gone. They say – oh Eichmann, he was nothing, he just arranged the train schedules. But it turns out that they took a scene from here and voiceover there. They have Eichmann giving testimony and they show Landau arranging his books – but in reality it didn’t happen together.

But the truly annoying thing is how many people come up to me and say “Oh, I read about the trial, I know about the trial, I read “Eichmann in Jerusalem” [Arendt’s book]. That really annoys me.

“Holocaust abuse” and Jewish hysteria

Q. Let me take you up on that. You say that it’s a question of degree and that Israelis for the first time could identify with the survivors. So, now, 50 years have gone by and I ask you to consider the possibility that that Jews, Israelis specifically, may be taking identification with Holocaust victims too far, and integrating this identification into their daily lives.

Lipstadt: Yes, people say “we remember the Shoah and we won’t let this happen again.”

First of all – I’ll tell you when it goes too far. When you say that because of anti-Semitism I’m going to be Jewish, or because of the Holocaust I’m going to support Israel. It shouldn’t be “mipnei” – (because) – it should be “af-al-pi” – (despite). If you let your support for Israel or your Jewishness become the result of the anti-Semites then you’ve ceded to the oppressor power over your identity. If the anti-Semites aren’t out there, I won’t do anything, I’ll go eat chazzer (pork), but if they’re out there – I’ll be on the barricades. And you’re turning your Jewish identity, your Israeli identity, whichever way you want to define it – culturally, politically, socially, religiously, or all of the above – you turn it into a defensive mechanism, as opposed to something that can nurture you, and strengthen you, and enrich you and open new vistas of thinking about things. If anti-Semitism becomes the reason through which your Jewish view of the world is refracted, if it becomes your prism, then it is very unhealthy. Jewish tradition never wanted that. It’s not because of, it’s in spite of.

Q. But what about the political use of the Shoah?

Lipstadt: And it has also become political. It’s a use and abuse of the Shoah. It’s a politicization of the Shoah. That doesn’t mean there aren’t political lessons to be learned from the Shoah – from anything – but it’s a use and abuse that I think is dangerous, just plain dangerous. Not only dangerous, because that can be debated, it’s a distortion of what Israel is all about, what Zionism is all about.

You listen to Newt Gingrich talking about the Palestinians as an “invented people” (there was a letter in the Times today saying that if the Palestinians are an invented people, please explain what are the Americans…). It’s out-Aipacking AIPAC, it’s out Israeling Israel, it means you’re against a two-state solution. Even Bibi Netanyahu is for a two-state solution, – I don’t think he really is, but he says he’s for a two-state solution.

I remember when Hillary Clinton spoke about a Palestinian state [in 1998] – the whole world went crazy. People go nuts here, they go nuts. I have friends in the Orthodox world, in the AIPAC world, they say I’m a leftie, and others who say I’m a rightie, because I want to call each side out and say – you’re wrong, you’re overstating the case. There’s no nuance, there’s no middle ground, it’s taking any shade of grey and stomping on it – and it’s dangerous, for your support of Israel to become a litmus test. It’s not for the two per cent of the US population who are Jews – they vote more, so maybe it’s equivalent to 6 per cent – it’s the Evangelicals. If you get the hechsher (kosher seal of approval) on this, you’re going to get a million Evangelicals to vote with you.

And it’s embarrassing. As a Jew who spends a lot of time in Israel, who could have lived in Israel if my life, you know, who happily identifies as a Jew – when I hear that, I cringe. We’re not a basket case. There’s something about it that’s so discomforting, it’s not healthy, it’s a distortion, This is the kind of thing that scares me: we’ve always been neurotic – I mean everyone’s neurotic, we just recognize it more – but we’ve raised our neuroses to a level that’s not healthy. We should eschew hysteria, but we don’t. Hysteria never is useful.

Someone told me that at the last AIPAC convention in May there was a group of Israeli officers and they said afterwards that it’s amazing to see all of this support for Israel. But, they asked – what on earth are these Jews so angry about? America is with us – what are they so angry about?

And there’s Ahmadinejad, I mean he’s a dangerous man, he’s a hater of Jews, if he had his hands on a nuclear weapon it would not be a good thing. But I question people about making too much of a fuss about Ahmadinejad, because one day the Ayatollahs are going to get rid of him, and then what – that solves our problem? From many points of view, including his Holocaust denial, Ahmadinejad is the gift that keeps on giving.

I was speaking to Saul Friedlander a while ago, and a really smart guy came to us and he said – do you feel that the situation is Europe today is similar to 1939? And after Saul answered him I said to Saul – this guy, he doesn’t what the Shoah is or he doesn’t understand what the situation in Europe is? I mean don’t get me wrong – the situation in Europe is not good, I wouldn’t want to send my kids to Jewish school in Belgium.

Q. What about the use of the word “appeasement” in the Republican debates and the use of the Holocaust in describing Israel’s present situation ? Is this a form of Holocaust denial?

Lipstadt: I wouldn’t call it that. I would call it a form of Holocaust abuse or instrumentalization of the Holocaust. That you take these terrible moments in our history, moments that deserve to be treated truthfully, and exactly, without exaggeration, in which the facts should speak for themselves. And you use it for contemporary purposes, and in so doing, in order to fulfill your political objectives, you mangle history, you trample on it. You’re doing that is akin to what the deniers do. I call it soft-core denial. It’s not the David Irvings, who deny the existence of the gas chambers. The soft deniers are those who say “the genocide of the Palestinians” or ‘the Nazi-like tactics of Zahal.”

Q. What about the settler youth, who call the IDF soldiers Nazis?
Lipstadt: That’s another example. It’s despicable. And it’s so inaccurate. And it’s such an abuse of history. The people who start it know it’s not true, but the kids, the yeshiva kids, and the high school kids – they don’t know it’s not true. And so when real Nazism comes around no one will recognize it.

One of the results of dealing with deniers has been my absolute devotion to truth. I was at Yeshiva University and I mentioned the story, that’s been thoroughly discredited by now, which was spread by a Beis Yaakov rabbi of the 93 Beis Yaakov girls who committed suicide in Cracow rather than be raped by the Nazis. And one of the girls supposedly got a letter out to New York. Now you tell me how in 1943 a letter made its way from Cracow to New York. Even if you were living in Paris you couldn’t get a letter out. And I said – it doesn’t make sense. You don’t need this story. It’s as if to say that what they went through wasn’t bad enough.

Ambassador Gutman’s remarks on the “new” anti-Semitism

Q. What is your opinion of what Ambassador Gutman said about the lack of resolution to the conflict fueling Muslim anti-Semitism?

Lipstadt: What he said was stupid. He was trying to be a political analyst, when he is an ambassador. He sounded as if he was rationalizing anti-Semitism. Anti-Semitism is a form of prejudice: if I see a black man walking down the street I think he’s lazy, if I see a Jew, I think he’s rich, if I see a pretty blond woman, I think she’s stupid. Prejudice is irrational. When you make a claim that Israelis are obnoxious, I mean we know some Israelis are obnoxious, but many are not. The minute you try to pin a rational explanation for prejudice, you say oh, it’s got a legitimate reason.

I’ll give you an example, a number of years ago I was giving a course about film in the Holocaust, together with a colleague who was an expert on film and religion and history. We agreed that we wouldn’t interrupt each other. So this was a week when they were seeing anti-Semitic propaganda films, “The Eternal Jew” and “Jud Suss” and so on, and so I’m talking about anti-Semitism, I’m talking about the irrationality of anti-Semitism and a student gets up and says – but weren’t all the bankers and lawyers in Germany Jews? So I said – there were 600,000 Jews in Germany, half of them were women, there were a whole bunch of children and I’m going on and on and suddenly my colleague, a former nun named Barbara DeConcini, a good Italian Catholic from Philadelphia, she suddenly interrupts me and she looks at the student and she says: “So what?” That was the right answer – So what? – and I was giving the wrong answer.

So I think that Gutman was trying to explain it. I think that some of these Muslims would hate Jews nonetheless, but the situation is definitely exacerbated by the conflict. I don’t think every Muslim hates Jews.

Is there a new anti-Semitism? In Europe and America, it comes more from the left than from the right. But a lot of it is the same. The situation in Israel gives the anti-Semites an opening, and then there are those people who aren’t necessarily anti-Semites but are willing to demonize Israel, as it says in the Finkler Question “this Jew discovered she was a Jew when she began to hate what Israel was doing the Middle East.” Is she an anti-Semite? No. But she buys into that picture.

But I also think the reaction to Gutman has been over the top, but that’s with everything, like we said before, with AIPAC, with everything. I think this kind of hysteria plays into it.

But you know there are no voices of calm, there are no voices of reason, not in this country, not in Israel. Obama was so flat-footed in the beginning you know like one Israeli said: you’re going to Cairo, you’re in the neighborhood and you don’t even come in for a cup of coffee? You stress the settlements, the settlements the settlements without making it a more even-handed kind of thing?

He just gave an opening to Republicans in America and to “Republicans” in Israel. I mean, it’s embarrassing. More and more Jews who are scared and here’s someone who is going to protect them. It’s so over the top irrational.

I teach in Queens College and a woman comes up to me and she says “first of all I love your book, I’m reading the Eichmann book” – so of course she’s an intelligent woman – and then she says: But I worry, aren’t we facing another Holocaust now? You’re in Queens, she said, and last year there were three cars that were torched near my synagogue and a swastika was painted on a doorstep. So I said – wait a minute, let’s talk about Rwanda, about Darfur, about the former Yugoslavia – do you see a government supporting that torching? Do you not see the police investigating that torching? So she just looked at me and said: “You don’t understand. Someone who’s written such a good book, but you don’t understand.”

Q. So that’s where we started, with the legacy of the Eichmann trial.

A. I think the legacy of the Eichmann trial is more positive than that. The fact that Israel could have killed him, and killed him in a way that everyone would know that Israel did it, but Ben Gurion said to bring him in. And [Judge] Moshe Landau tried to run a fair trial, which was difficult under the circumstances. So I think it was important, here were the victims who could have just thrown this guy overboard, trying to do a modicum of justice, trying to prove his guilt – it was clear that he was going to be found guilty, but really feeling obligated to prove it.

I don’t think what we’re seeing now is a legacy of the trial. I think what we’re seeing now is that there is a hysteria, a fear, certainly born out of real concern, born out of politics, it’s possibly a hysteria that is buttressed by a certain racism, that is leading people to instrumentalize the Holocaust, to instrumentalize anti-Semitism in the way of this pandering appeal at the Republican Jewish Coalition. It was unbelievable. I couldn’t watch it. And people say to me – what? Look how important we are. What’s wrong with you?

 

Russian Authorities Pressure Elections Watchdog, by Ellen Barry

MOSCOW — With parliamentary elections only days away that are expected to reflect dwindling public support for Vladimir V. Putin’s party, Russian prosecutors have opened a case against the country’s only independent election monitoring organization.

Grigory A. Melkonyants, the deputy director of Golos, Russia’s only independent election monitoring organization, said he was worried that the group’s 3,000 monitors would be excluded from polling stations.

The organization, Golos, has already posted reports of more than 4,500 violations of election law in the prelude to the voting on Sunday. Golos receives financing from Western governments, including the United States, and some Russian officials have suggested that the organization’s real aim is to incite an Arab Spring-type revolution in Russia.

Pressure on Golos, an 11-year-old group whose name means “vote” in Russian, began mounting last Sunday, when Mr. Putin attacked “so-called grant recipients” that he said were interfering with elections on behalf of foreign governments. “Judas,” he said, “is not the most respected biblical figure among our people.”

The accusations became specific this week, when lawmakers from three parties appealed to the federal prosecutor to investigate Golos. Blistering attacks appeared in several pro-government news outlets. In one article, a leader of Mr. Putin’s party, United Russia, likened Golos to a group announcing that “we got money from Al Qaeda, and now we want to see how elections in Tatarstan are going.”

The mood at Golos’s Moscow office was jumpy on Thursday. Grigory A. Melkonyants, the organization’s deputy director, said he was worried that the group’s 3,000 monitors would be excluded from polling stations, and that they might even be in danger, after Mr. Putin’s use of the word “Judas.” He said it was possible that Golos would be shut down by the authorities before Sunday.

“We cannot predict what will happen tomorrow, or what will happen in an hour,” Mr. Melkonyants said. “This is already an open war that they have declared on us, and the overwhelming weight is on their side.”

A letter delivered by prosecutors on Thursday said that one of Golos’s most popular features — an online “map of violations” where people can post reports — violates a Russian law against publishing data, especially polling results, during the five days before voting. This is an administrative violation punishable by a fine of up to 100,000 rubles, or about $3,200.

The letter also accuses Golos, which has been preparing for Sunday’s election for years, of “dissemination of rumors under the guise of trustworthy reports, with the goal of defaming a party as well as its individual members.”

Though United Russia, which now has a commanding majority in Parliament, faces no powerful competitors in the election, opinion polls suggest that it will lose 50 to 60 seats, reflecting growing weariness with leadership that has not changed in a decade. State officials at all levels have been told to guard against significant losses.

Meanwhile, amateur observers are pushing back, using the Internet to expose violations that include offers of cash and threats to cut off financing. Dmitri Merezhko, Golos’s communications director, said that events this fall — including the collapse of a pro-business party, Right Cause, and Mr. Putin’s announcement that he will run for president again, after a stint as prime minister and two previous terms as president — “ignited something in the most active part of the public.”

“Maybe someone thinks we are the reason,” Mr. Merezhko said, referring to Golos. “But we are not the reason.”

Golos’s critics in the Russian government say its work is tainted by the money it receives from two American agencies, the National Endowment for Democracy and the United States Agency for International Development. A promotional video clip for a report scheduled to be broadcast on Friday on the NTV channel, owned by the Russian energy giant Gazprom, features images of suitcases stuffed with $100 bills juxtaposed with footage of Golos’s leaders as a portentous voice asks, “Who is behind these ‘independent observers?’ ” A pro-government blogger has posted what appears to be paperwork showing that Golos received $92,653 from the United States government for the month of February.

Russian officials have invoked the memory of the 2004 Orange Revolution in Ukraine, which many people here believe was propelled by American agents. Maksim Rokhmistrov, one of the three lawmakers who signed the appeal to the prosecutor, said he was worried that Golos’s reporting could provoke civil unrest.

“A girl dances with the guy who takes her to dinner, as they say,” Vzglyad, a daily newspaper, quoted Mr. Rokhmistrov as saying this week. “The question of maintaining stability in the state worries everyone at election time. No one here wants to see the beginning of our own Egypt, or our own Libya.”

Mr. Melkonyants has given a number of interviews with the news media since the critiques began to appear, arguing that the foreign financing does not compromise Golos’s objectivity or violate any Russian law. He told the radio station Ekho Moskvy that much of Golos’s money came from European donors like the Helsinki Committee and the European Commission.

He said Golos had submitted to rigorous audits by Russian tax authorities and the Ministry of Justice, and had repeatedly tried to secure financing from the Russian government but had been refused.

Mr. Melkonyants is scheduled to appear in a Moscow courtroom on Friday at the request of prosecutors.

Though every election monitoring mission has come under pressure from government officials, “it has never been so high in my memory,” he said, adding that he could not blame Golos’s partners for keeping their distance now. “There is no need to say that they are cowards,” he said. “Believe me, all of us — including the political parties — we’re all in a situation that is not simple, not simple at all.”

 

A simple proof of the Mattila-Sjolin theorem

The Mattila-Sjolin theorem says that if the Hausdorff dimension of is greater than , the the distance set contains an interval. Mihalis Mourgoglou, Krystal Taylor and recently extended this result to distance sets , where is the distance generated by a symmetric convex body with a smooth boundary and everywhere non-vanishing Gaussian curvature. We also prove other things, but my goal here is to describe our proof of the Mattila-Sjolin result extended to smooth well-curved metrics.

Define the measure on by the relation

, where is a Frostman measure on . We prove that

, where is continuous away from the origin and . Indeed, in place of  we may consider

, where is the Lebesgue measure on and is convolved with the approximation to the identity at level , i.e .

By Plancherel, this expression equals

.

The function is continuous away fro by the dominated convergence theorem since

by the method of stationary phase and

since is a Frostman measure on of Hausdorff dimension greater than .

It remains to show that . But

and this quantity goes to as if .

This argument shows that can be replaced by a much more general function.

Recruiting trip to Ukraine, Day 4-5

The fourth day ended very nicely. At 4 p.m., Vladimir Kadets picked me up and took me to his house, a short metro ride away from the University. There I met his wife and his son, both pleasant and interesting people. We spent a very nice evening together discussing mathematics, academic interactions, educational issues and world politics. At the end of the dinner, Vladimir presented me with a copy of his functional analysis book. I spent quite a bit of time looking at the book since and I enjoyed it greatly. It should prove a good source of additional material and problems for my measure theory course this semester.

The evening began in a somewhat funny way when I realized that I did not tell the Kadets family that I am allergic to chicken. The first and the second course they prepared contained poultry, but they quickly regrouped and put a wonderful dinner on the table consisting of cheese, sausages, eggplant and mushroom pastries. I also inquired if I can buy some pork lard at the store nearby, but when we got there, there was none left as the store was about to close. Oh well. I will have to make do with the pork lard the Russian store in Rochester.

It is now 3:19 a.m. on the fifth day of the trip. My taxi is due to arrive at 4:45 and my flight to Kiev is at 7 a.m.. After spending a few hours in Kiev, I am due to fly to Paris and then on to New York, JFK and then to Rochester. I should arrive there at 10:47 p.m. tonight and then I have to teach on Monday. And I have not yet prepared for my measure theory class, so it will be an interesting Monday morning, afternoon and evening. I say evening because my calculus students have a test on Tuesday, so I plan to run a review on Monday night.

I am now sitting at the 24 hour restaurant in Hotel Kharkov, waiting for my crepes with red caviar to arrive. The prices in this country are still quite reasonable, which allows for a very nice lifestyle during the visit. I am definitely planning to come back in the near future. This has been a fantastic and potentially successful visit!

Recruiting trip to Ukraine, Day 4 (things get slightly humorous)

This was bound to happen sooner or later and it finally occurred early on the fourth day of my trip. I felt a sudden urge to laugh uncontrollably. At exactly 7 a.m. I went to the breakfast room on the 8th floor of Hotel Kharkov where I am staying. This is a rather comfortable four star hotel with a polite staff and all the reasonable amenities. I skipped breakfast yesterday because I decided to sleep in, but today I was exposed to the full humor of the situation. I was served a small glass of orange juice, a tiny plate with cheese and salami and then I was given a choice of sausages OR eggs. I could not have both. What is more amusing is that I could not even pay to have both unless I went to the restaurant on the ground floor. This was not a big deal because I was given plenty of food.

The funny part came later when I was offered tea or coffee. I chose tea and enjoyed a very nice cup of black tea with lemon. After I asked for a second cup, I was told that the limit is one cup of tea per customer. The English tourists were clamoring about that in the neighboring room, but I had assumed that they were joking. But no, one cup per customer is the deal and, you guessed it, the only way to get more is to go to the restaurant on the ground floor.

I am now back in my room wondering how the managerial staff has not figured out that this is not the way to turn a profit in a four star hotel. But then again, I am certainly not a businessman, so I perhaps I am missing something. On the positive side, missing out on that second cup of tea should allow me to take a short post-breakfast nap that I badly need before my upcoming stroll around the city. Life is good! And funny!

Recruiting trip to Ukraine, Day 3

The past few days have been quick paced and interesting beyond all my expectations. I met with three different groups of students at the Taras Shevchenko University in Kiev and another large group at the University of Kharkov. All the meetings were interesting, lively and full of interesting questions. I was impressed with the politeness and intelligence emanating from the students and the faculty I met. The hospitality I received from Oksana Bezuschak and other faculty in the department was also very impressive.

The differences between the American and Ukrainian educational systems came up repeatedly. The conclusion seems to be that while both systems are deteriorating, the Ukrainian system started falling from a higher cliff, so they are still better off than we are. It was quite apparent that many students had to work extensively while taking classes and this gives me hope that the prospect of receiving good financial support and limited teaching duties at the graduate program at the University of Rochester would be appealing to them.

I have already received several emails from the students I spoke with, which gives me hope that several strong applications will result from this visit. But work aside, it struck me throughout the visit how comfortably I felt dealing with people at the hotel, the university and on the street. All the small fears I experienced on the way to visiting the country of my birth vanished into this air a few hours after I arrived. There are issues here in Ukraine, to be sure, but I definitely look forward to returning here in the near future and exploring the country more thoroughly. In particular, I now feel ready to visit Lvov, my hometown.

Recruiting trip to Ukraine, Day 2

After doing a tiny bit of sight seeing last night, I went to sleep and kept waking up every couple of hours, doing work, watching TV, browsing the internet, talking to Shannon and the little honeys on Skype, shaving, showering and gathering my thoughts for the upcoming trip to the Taras Shevchenko University mathematics department later this morning. I have not been to this country in 32 years and I really do not know what to expect. Do the students know about graduate programs in the United States? Are they remotely interested in moving so far away? I guess there is only one way to find out. And history has shown that many students before them did make this journey, so my efforts are at the very least not entirely unreasonable.

I am also thinking about the best way to present the mathematics department at Rochester. If they could spend a day there, they would understand how wonderful it is to be a faculty or a student there, but describing this phenomenon from thousands of miles away will be a bit of a challenge. How do I explain that students are treated like human beings and that people treat each other with respect and kindness that is less and less frequent in academic institutions around the world? How do I explain that being a teaching assistant for calculus in Rochester is not the same as it is in a typical state university? I guess the best way is to just describe things exactly the way I feel about them and go from there.

I am finishing up breakfast and the experience has its nuances. The food is quite good, but the coffee is terrible and tea dispenser is not working. It is precisely this sort of an attention to detail that separates this hotel from similar operations in Northern Italy, for example. The hotel attendants are much more polite than I what I recall from the Soviet era, but several of them have retain some of their old habits.

The police is exactly as I remember them from the late 70s. I went out last night to find a pharmacy and asked a couple of cops if there is one in the neighborhood. One of them started explaining to me that he is not a tour guide, while the other, the nice one of the bunch, pointed in a vague direction and told me that it is 400 meters away. But overall the experience has been quite pleasant and I am looking forward to the rest of my trip with great anticipation.

Recruiting trip to Ukraine, Day 1

I am back in Ukraine for the first time in 32 years. The experience so far is somewhat anti-climactic, not that I expected any great emotional upheaval. The flight was relatively uneventful, with somewhat excessive turbulence on the flight from Atlanta to Paris and less than exciting food throughout. The movies were predictably cheesy and the service less than completely enthusiastic, but on the plus side people sitting next to me were thoroughly reasonable. Nobody bugged me about about the endless yellow sheets of paper filled with formulas and I did not receive a single comment about how impossible mathematics is. If they only knew…

Upon arriving in Kiev, I booked a hotel room in the center of the city. The prices are more reasonable than I was led to believe and the general demeanor has improved significantly from the Soviet days. The hotel attendant was outright polite and the taxi driver, named Ruslan, did a very nice impersonation of a tour guide on the way from the airport to the city center. I am now comfortably situated in my room, struggling between taking a short nap and going out to see the city immediately.

Most of the people in the airport and in the hotel are speaking Russian. I have not heard any Ukrainian speech yet, though all the signs on the streets are in Ukrainian. I was not sure what to expect in this regard. Most of my information about Ukraine comes either from Kharkov, where Russian is spoken almost exclusively, and from Lvov, where Ukrainian is fully dominant. Kiev appears to be mainly Russian speaking, though the airport and the hotel may not be the best indicators. We shall see what the rest of the day brings.

It is time to get some lunch! I plan to be online later in the day with more impressions of my journey. Tomorrow I meet with the University of Kiev folks in hopes of recruiting some potential graduate students for the University of Rochester mathematics department. There will be much to write after that experience, I suspect.