Sums, Differences and Dilates

Document Type

Presentation Abstract

Presentation Date

4-6-2026

Abstract

Given a set A of integers, the sumset A + A is defined as A + A = {a + b : a, b ∈, A}, and the difference set A − A is defined as A − A = {a − b : a, b ∈ A}. The systematic study of sumsets was initiated in the 1960s by the Russian mathematician G A Freiman, who was motivated by several problems in number theory. Freiman proved a very influential inverse theorem for sumsets, now named after him. But he also, together with Pigarev, proved that if A is finite, then |A + A|3/4 ≤ |A − A| ≤ |A + A|4/3. This bound was later improved by Imre Ruzsa. Meanwhile, in 1967, J H Conway constructed a surprising set A of size 8 for which |A + A| = 26 > 25 = |A − A|.


These results all lead to the question of bounding |A − A| in terms of |A| and |A + A|. So we ask: what the are possible values (x, y) such that there exists a set A with |A + A|=|A|x and |A − A| = |A|y? The closure of the set of such points is the feasible region F1,−1. Using a probabilistic technique of Ruzsa, a geometric construction of Hennecart, Robert and Yudin, methods from information theory, and the (new?) concept of the size of a fractional dilate, we prove that the point (1.7354, 2) is feasible (i.e., contained in F1,−1). Previously, no explicit point (c, 2) with c < 2 was known to be feasible. We also present some new results about the size of the dilate set A + 2 ⋅ A = {a + 2b : a, b ∈ A}, and its corresponding feasible region F1,2.

This is joint work with Jon Cutler and Luke Pebody.

Additional Details

April 6, 2026 at 3:00 p.m. Math 103

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