Higher convexity and iterated sum sets

Abstract: An important generalisation of the sum-product phenomenon is the basic idea that convex functions destroy additive structure. This idea has perhaps been most notably quantified in the work of Elekes, Nathanson and Ruzsa, in which they used incidence geometry to prove that at least one of the sets $A+A$ or $f(A)+f(A)$ must be large.

I will discuss joint work with Hanson and Rudnev, in which we use a stronger notion of convexity to make further progress. In particular, we show that, if $A+A$ is sufficiently small and $f$ satisfies this hyperconvexity condition, then we have unbounded growth for sums of $f(A)$. This in turn gives new results for iterated product sets of a set with small sum set.

Title: An update on the state-of-the-art sum-product inequality over the reals Abstract: The aim of this somewhat technical talk is to clarify the underlying constructions and present a streamlined step-by-step self-contained proof of the sum-product inequality of Solymosi, Konyagin and Shkredov. The proof ends up with a slightly better exponent $4/3+2/1167$ than the previous world record.

Joint work with Sophie Stevens.

number theory

Audience: researchers in the topic

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Combinatorial and additive number theory (CANT 2021)

Series comments: This is the nineteenth in a series of annual workshops sponsored by the New York Number Theory Seminar on problems in combinatorial and additive number theory and related parts of mathematics.

Registration for the conference is free. Register at cant2021.eventbrite.com.

The conference website is www.theoryofnumbers.com/cant/ Lectures will be broadcast on Zoom. The Zoom login will be emailed daily to everyone who has registered on eventbrite. To join the meeting, you may need to download the free software from www.zoom.us.

The conference program, list of speakers, and abstracts are posted on the external website.

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