Lewko's blog

Adam Smith on mathematicians (circa 1759)

Posted in Uncategorized by Mark Lewko on January 29, 2011

I recently came across the following passage regarding the mathematical profession from Adam Smith’s  influential work The Theory of Moral Sentiments that I thought others might find interesting:

   Mathematicians, on the contrary, who may have the most perfect assurance, both of the truth and of the importance of their discoveries, are frequently very indifferent about the reception which they may meet with from the public. The two greatest mathematicians that I ever have had the honour to be known to, and, I believe, the two greatest that have lived in my time, Dr Robert Simpson of Glasgow, and Dr Matthew Stewart of Edinburgh, never seemed to feel even the slightest uneasiness from the neglect with which the ignorance of the public received some of their most valuable works. The great work of Sir Isaac Newton, his Mathematical Principles of Natural Philosophy, I have been told, was for several years neglected by the public. The tranquillity of that great man, it is probable, never suffered, upon that account, the interruption of a single quarter of an hour. Natural philosophers, in their independency upon the public opinion, approach nearly to mathematicians, and, in their judgments concerning the merit of their own discoveries and observations, enjoy some degree of the same security and tranquillity.

   The morals of those different classes of men of letters are, perhaps, sometimes somewhat affected by this very great difference in their situation with regard to the public.

   Mathematicians and natural philosophers, from their independency upon the public opinion, have little temptation to form themselves into factions and cabals, either for the support of their own reputation, or for the depression of that of their rivals. They are almost always men of the most amiable simplicity of manners, who live in good harmony with one another, are the friends of one another’s reputation, enter into no intrigue in order to secure the public applause, but are pleased when their works are approved of, without being either much vexed or very angry when they are neglected.

The entire text is available here.

An improved upper bound for the sum-free subset constant

Posted in math.CO, Paper, Uncategorized by Mark Lewko on August 27, 2010

I recently arXiv’ed a short note titled An Improved Upper Bound for the Sum-free Subset Constant. In this post I will briefly describe the result.

We say a set of natural numbers A is sum-free if there is no solution to the equation x+y=z with x,y,z \in A. The following is a well-known theorem of Erdős.

Theorem Let A be a finite set of natural numbers. There exists a sum-free subset S \subseteq A such that |S| \geq \frac{1}{3}|A|.

The proof of this theorem is a common example of the probabilistic method and appears in many textbooks. Alon and Kleitman have observed that Erdős’ argument essentially gives the theorem with the slightly stronger conclusion |S| \geq \frac{|A|+1}{3}. Bourgain  has improved this further, showing that the conclusion can be strengthened to |S| \geq \frac{|A| + 2}{3}. Bourgain’s estimate is sharp for small sets, and improving it for larger sets seems to be a difficult problem. There has also been interest in establishing upper bounds for the problem. It seems likely that the constant 1/3 cannot be replaced by a larger constant, however this is an open problem. In Erdős’ 1965 paper, he showed that the constant \frac{1}{3} could not be replaced by a number greater than 3/7 \approx .429 by considering the set \{2,3,4,5,6,8,10\}. In 1990, Alon and Kleitman improved this to 12/29 \approx .414. In a recent survey of open problems in combinatorics, it is reported that Malouf has shown the constant cannot be greater than 4/10 = .4.  While we have not seen Malouf’s proof, we note that this can be established by considering the set \{1,2,3,4,5,6,8,9,10,18\}. In this note we further improve on these results by showing that the optimal constant cannot be greater than 11/28 \approx .393.

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Halstead has a home!

Posted in Uncategorized by Mark Lewko on November 21, 2009

Two weeks ago, not far from the UT math department, I found (or rather I was found by) a very friendly stray dog (pictured below). Since it was raining and the nearby streets were busy, I fed the dog and then brought it to Austin’s Town Lake animal shelter. The next day I called the shelter to learn that if the dog wasn’t adopted within three days it would likely be euthanized. With the help of several other members of the UT mathematical community, dozens of emails, phone calls, and Internet postings were made in an effort to find the pup a home. (In fact, the mathematical blogsphere was represented in these efforts.) (more…)

Thin sets of primes and the Goldbach property

Posted in Fourier Analysis, math.NT, Uncategorized by Mark Lewko on November 19, 2009

As is well known, Goldbach conjectured that every even positive integer (greater than 2) is the sum of two primes. While this is a difficult open problem, progress has been made from a number of different directions. Perhaps most notably, Chen has shown that every sufficiently large even positive integer is the sum of a prime and an almost prime (that is an integer that is a product of at most two primes). In another direction, Montgomery and Vaughan have shown that if {E} is the set of positive even integers that cannot be expressed as the sum of two primes then

 \displaystyle |[0,N]\cap E| \leq |N|^{1-\delta}

for some positive constant {\delta>0}. This is stronger than the observation (which was made much earlier) that almost every positive integer can be expressed as the sum of two primes. In this post we’ll be interested in sets of integers with the property that most integers can be expressed as the sum of two elements from the set. To be more precise we’ll say that a set of positive integers {S} has the Goldbach property (GP) if the sumset {S+S} consists of a positive proportion of the integers. From the preceding discussion we have that the set of primes has the GP. (This discussion is closely related to the theory of thin bases.)

A natural first question in investigating such sets would be to ask how thin such a set can be. Simply considering the number of possible distinct sums, the reader can easily verify that a set {S} (of positive integers) with the GP must satisfy

 \displaystyle \liminf_{N\rightarrow\infty} \frac{[0,N]\cap S}{\sqrt{N}} > 0.

This is to say that a set of positive integers with the GP must satisfy {[0,N]\cap S \gg \sqrt{N}} for all large {N}. Recall that the prime number theorem gives us that {|[0,N]\cap\mathcal{P}| \approx N/\ln(N)} for the set of primes {\mathcal{P}}. Thus the primes are much thicker than a set with the GP needs to be (at least from naive combinatorial considerations).

Considering this, one might ask if there is a subset of the primes with the GP but having significantly lower density in the integers. I recently (re)discovered that the answer to this question is yes. In particular we have that

 Theorem 1 There exists a subset of the primes {\mathcal{Q}} such that {\mathcal{Q}+\mathcal{Q}} has positive density in the integers and

\displaystyle \limsup_{N\rightarrow\infty} \frac{[0,N]\cap \mathcal{Q}}{\sqrt{N}} < \infty.

(more…)

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The sharp large sieve inquality

Posted in expository, Fourier Analysis, math.NT, Uncategorized by Mark Lewko on August 31, 2009

This is the first post in a short sequence related to the large sieve inequality and its applications. In this post I will give a proof of the sharp (analytic) large sieve inequality. While the main result of this post is a purely analytic statement, one can quickly obtain a large number of arithmetic consequences from it. The most famous of these is probably a theorem of Brun which states that the sum of the reciprocals of the twin primes converges. I will present this and other arithmetic applications in a following post. This post will focus on proving the sharp (analytic) large sieve inequality. Much of the work in this post could be simplified if we were willing to settle for a less than optimal constant in the inequality. This would have little impact on the the arithmetic applications, but we’ll stick to proving the sharp form of this inequality (mostly for my own benefit). I should point out that there is an alternate approach to this inequality, via Selberg-Buerling extremal functions, which gives an even sharper result.  In particular, using this method one can replace the N+\delta^{-1} in Theorem 1 below by N-1+\delta^{-1}.  We will not pursue this further here, however.  Much of this post will follow this paper of Montgomery, however I have made a few attempts at simplifying the  presentation, most notably I have eliminated the need to use properties of skew-hermitian forms in the proof of the sharp discrete Hilbert transform inequality.

(more…)

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