Misha Rudnev

University of Bristol

On convexity and sumsets

Heilbronn Number Theory Seminar

4th November 2020, 4:00 pm – 5:00 pm

A finite set A of n reals is convex if the sequence of neighbouring differences is strictly monotone. Erdös suggested that that the set of squares of the first n integers may constitute the extremal case, namely that for any convex A, |A+A| > n^{2-o(1)}. The question is still open, we'll review some partial progress.

What about k- fold sums A+A+A+...? In the case of squares, they stop growing after k=2, and for kth powers they grow up to n^k. In a joint work with Brandon Hanson and Olly Roche-Newton we show, using elementary methods, that if A=f([n]), where f is a real function with k-1 strictly monotone derivatives, taking sufficiently many sums does lead to growth up to n^{k-o(1)}. We generalise this by replacing the interval [n] versus f([n]) by any set with small additive doubling versus its image by f, which enables us to apply this to sum-product type questions.

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