On Sumsets of Convex Sets

Abstract

A set of reals A = {a 1 , . . . , a n } is called convex if a i+1 − a i > a i − a i−1 for all i. We prove, among other results, that for some c > 0 every convex A satisfies |A − A| c|A| 8/5 log −2/5 |A|.

Convex sequences may have thin additive bases

“…The first non-trivial bound confirming the conjecture of Erdős was obtained by Hegyvári [2], and the state of the art bound is due to Schoen and Shkredov [3], who proved that for an arbitrary convex set A holds |A + A| ≥ C|A| 14/9 log −2/3 |A| for some absolute constants C, c > 0. It is conjectured that in fact |A + A| ≥ C(ǫ)|A| 2−ǫ holds for any ǫ > 0.…”

On sums of Szemerédi-Trotter sets

“…Applying the first formula of Lemma 7 to estimate the quantity E 3 (A) and Lemma 10 to estimate We conclude the section by proving a result that generalizes, in particular, Theorem 1.3 from [9] as well as the results on sumsets/difference sets of convex sets from [12]. The arguments are in the spirit of Theorem 11.…”

“…Using Lemma 13, we prove our second main result, although one can use a more elementary approach as in [12]. and for α > 1 Proof.…”

“…Using Definition 5 and easy calculations, one can obtain upper bounds for some simple characteristics of SzT-type sets (see, e.g., [8,9,12,13]). It is more convenient to use c (A, B) instead of the parameter α in the following lemma.…”

Additive Combinatorics: With a View Towards Computer Science and Cryptography—An Exposition

“…The sum-product phenomenon says that if a finite set A is not close to a subring, then either the sumset A + A or the product set A · A must be considerably larger than A. The reader can refer to [276] and the references therein to see some lower bounds on |C − C| and |C + C|, where C is a convex set (a set of reals C = {c 1 , . .…”