Suppose that A is a finite, non-empty subset of a cyclic group of either infinite or prime order. We show that if the difference set A-A is 'not too large', then there is a non-zero group element with at least as many as (2+o(1))|A|2/|A-A| representations as a difference of two elements of A; that is, the second largest number of representations is, essentially, twice the average. Here the coefficient 2 is best possible.
For a subset A of a finite-dimensional vector space over a finite field, It is known that 2A-2A must contain a subgroup of density that is quasi-polynomial in the density of A. We show an analogue result for the special linear group over a finite field, except we achieve a polynomial density for the subgroup. The main tool used is a new hyper-contractive estimate for the special linear group, that has origins in the study of 2-to-2 games.
A central question in additive combinatorics is to understand how large arithmetic progression-free sets can be. In this talk, I will focus on this question for high-dimensional generalization of arithmetic progressions (AP) known as corners. A (2-dimensional) corner is a triple of the form (x,y),(x+d,y),(x,y+d) for some d>0 in [N] × [N]. Extending this definition to higher dimensions, a k-dimensional corner in [N]k is a (k+1)-tuple defined similarly for some d. While it is known that corner-free sets have a vanishingly small density, the precise bounds on their size remain unknown. Until recently, the best-known corner-free sets were derived from constructions of AP-free sets: a construction of a 3-term AP-free set by Behrend from 1946, and a generalization by Rankin for k-term APs in 1961. New results by Linial and Shraibman (CCC 2021) and Green (New Zealand Journal of Mathematics 2021) changed this picture; they improved the upper bound for k=2 by adopting a communication complexity point of view.
I will discuss our recent work where we employ the same perspective of communication complexity and obtain the first improvement on the upper bound of the size of high-dimensional (k>2) corner-free sets since the original construction of Rankin.
Suppose that N is large and that A is a subset of {1,..,N} which does not contain two elements x, y with x - y equal to p-1, p a prime. Then A has cardinality at most N1 - c, for some absolute and positive c. I will discuss the history of this kind of question as well as some aspects of the proof of the stated result.
Ruzsa asked whether there exist Fourier-uniform subsets of ℤ/Nℤ with very few 4-term arithmetic progressions (4-AP). The standard pedagogical example of a Fourier-uniform set with a "wrong" density of 4-APs actually has 4-AP density much higher than random. Can it instead be much lower than random? Gowers constructed Fourier uniform sets with 4-AP density at most α4+c. It remains open whether a superpolynomial decay is possible. We will discuss this question and some variants. We relate it to an arithmetic Ramsey question: can one No(1)-color of [N] avoiding symmetrically-colored 4-APs?
We show that for every positive integer k there are positive constants C and c such that if A is a subset of {1, 2, ..., n} of size at least C n1/k, then, for some d ≤ k-1, the set of subset sums of A contains a homogeneous d-dimensional generalized arithmetic progression of size at least c|A|d+1. This strengthens a result of Szemerédi and Vu, who proved a similar statement without the homogeneity condition. As an application, we make progress on the Erdős-Straus non-averaging sets problem, showing that every subset A of {1, 2, ..., n} of size at least n√2 - 1 + o(1) contains an element which is the average of two or more other elements of A. This gives the first polynomial improvement on a result of Erdős and Sárközy from 1990.
It is an open question as to whether the prime numbers contain the sum A+B of two infinite sets of natural numbers A, B (although results of this type are known assuming the Hardy-Littlewood prime tuples conjecture). Using the Maynard sieve and the Bergelson intersectivity lemma, we show the weaker result that there exist two infinite sequences a1 < a2 < ... and b1 < b2 < ... such that ai + bj is prime for all i < j. Equivalently, the primes are not 'translation-finite' in the sense of Ruppert. As an application of these methods we show that the orbit closure of the primes is uncountable.
It is an open question as to whether the prime numbers contain the sum A+B of two infinite sets of natural numbers A, B (although results of this type are known assuming the Hardy-Littlewood prime tuples conjecture). Using the Maynard sieve and the Bergelson intersectivity lemma, we show the weaker result that there exist two infinite sequences a 1 < a 2 < ... and b 1 < b 2 < ... such that ai + bj is prime for all i < j. Equivalently, the primes are not "translation-finite" in the sense of Ruppert. As an application of these methods we show that the orbit closure of the primes is uncountable.
In a joint work with Daniel Fiorilli, we develop a new method to prove lower bounds of some moments related to the distribution of primes in arithmetic progressions. We shall present main results and explain some aspects of the proofs. To prove our results, we assume GRH but we succeed to avoid linearly independence on zeroes hypothesis.
We discuss some recent progress on the model-theoretic problem of classifying the reducts of the complex field (with named parameters and up to interdefinability). The tools we use include Castle’s recent solution of the Restricted Trichotomy Conjecture in characteristic 0 and a generalized sumproduct result from additive combinatorics.
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