Seminars in Additive Number Theory

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 N^{1 - 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.

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 n^1/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 a_i + b_j 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 a_i + b_j 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.