Growth in Sumsets of Higher Convex Functions

Abstract

The main results of this paper concern growth in sums of a k-convex function f . Firstly, we streamline the proof (from [9]) of a growth result for f (A) where A has small additive doubling, and improve the bound by removing logarithmic factors. The result yields an optimal bound forWe also generalise a recent result from [10], proving that for any finitewhere s = k+12 . This allows us to prove that, given any natural number s ∈ N, there exists m = m(s) such that if A ⊂ R, then(1)This is progress towards a conjecture [1] which states that (1) can be replaced withDeveloping methods of Solymosi, and Bloom and Jones, and using an idea from [5], we present some new sum-product type results in the complex numbers C and in the function field F q ((t −1 )).

Additive dimension and the growth of sets

“…To get (92) we take l n such that l kl < p/2. Then all primes up to l form a k-dissociated set Λ modulo p. The condition l kl < p/2 is equivalent to l ≪ log p/(k log log p) and by the prime number theorem we have |Λ| ≫ log p/k(log log p) The dependence on m in (90) is not logarithmic as in [10], [11] or [23] and the whole bound is much better than [41, Theorem 2]. As in [41] bound (90) is a step towards the main conjecture from [1], where authors do not assume that the additional condition of the doubling constant takes place.…”

Convexity, Squeezing, and the Elekes-Szabó Theorem

“…An important generalization of the sum-product phenomenon is the idea that strictly convex or concave functions destroy additive structure. 1 For instance, Elekes, Nathanson and Ruzsa [5] used incidence geometry to prove that the bound 5 holds for any A ⊂ R and any strictly convex function f .…”

“…A recent trend in this area of sum-product theory has seen elementary methods play a more prominent role. These elementary methods have their origins in the work of Ruzsa, Shakan, Solymosi and Szemerédi [17], who gave a simple and beautiful proof of the fact that (1) |A + A − A| ≫ |A| 2 holds for any convex set A ⊂ R. A set A is said to be (strictly) convex if it has strictly increasing consecutive differences. That is, if we write A = {a 1 < a 2 < • • • < a n } then a i − a i−1 < a i+1 − a i holds for all 2 ≤ i ≤ n − 1.…”

“…For the purposes of our work, we will apply an elementary "squeezing" method to two different number-theoretic problems. This method was the foundation for the proof of (1), and has already been further developed in a series of recent papers (see [1], [2], [8], [9], [13]).…”

“…The bound (3) is also tight, up to constant and logarithmic factors, as is illustrated by the case when A = [n] for some positive integer n and f (x) = x 2 . Moreover, a recent paper of Bradshaw [1] removed the logarithmic factors from (3).…”