The Uniformity Conjecture in Additive Combinatorics

Abstract

In this paper we show examples for applications of the Bombieri-Lang conjecture in additive combinatorics, giving bounds on the cardinality of sumsets of squares and higher powers of integers. Using similar methods we give bounds on the sum-product problem for matchings.

The Elekes-Szabó Problem and the Uniformity Conjecture

“…for any set A of rational squares, a result which first appeared in [7] and was recently improved slightly in [30]. These results are related to a conjecture of Rudin [26], which states that the number of squares in an arithmetic progression A is O(|A| 1/2 ).…”

“…It was conjectured by Chang [6] that the number of such solutions is bounded by |A| 2+ǫ for all ǫ > 0. Conditional results have recently been given by Shkredov and Solymosi [30], who improved on the aforementioned results of [7] and gave an optimal bound on the fourth order additive energy of a set of squares.…”

“…the number of edges in a certain graph) to rational points on some hyperelliptic curves. For instance, to bound the fourth order additive energy of a set of squares, Shkredov and Solymosi [30] used the genus 2 hyperelliptic curve y 2 = (x 2 +α)(x 2 +β)(x 2 +γ) for non-zero α, β, γ ∈ Z.…”

“…Alon, Angel, Benjamini, and Lubetzky [1] used the Uniformity Conjecture to give an improvement to (3). Further progress was obtained by Shkredov and Solymosi [30], still under the assumption of the Uniformity Conjecture. They focus on the case when G is a matching, but their proof implicitly gives the following more general bound for sums and products along graphs: given A, B ⊂ Q and G a bipartite graph on vertex set…”

Sums, products and dilates on sparse graphs

“…implicit in the work of Shkredov and Solymosi [13], holds for any A, B ⊂ Z and G ⊂ A × B. This builds on work of Alon et al [1].…”