Condensation and densification for sets of large diameter

Abstract: Consider a set of integers $A$ having finite diameter $X$, so that \[ \sup A-\inf A=X<\infty , \] and a system of simultaneous polynomial equations $P_1(\mathbf x)=\ldots =P_r(\mathbf x)=0$ to be solved with $\mathbf x\in A^s$. In many circumstances, one can show that the number $N(A;\mathbf P)$ of solutions of this system satisfies $N(A;\mathbf P)\ll X^\epsilon |A|^\theta$ for a suitable $\theta < s$ and any $\epsilon>0$. Such is the case with modern variants of Vinogradov's mean value theorem due to the author, and likewise Bourgain, Demeter and Guth. These estimates become worse than trivial when the diameter $X$ is very large compared to $|A|$, or equivalently, when the set $A$ is very sparse. This motivates the problem of seeking new sets of integers $A'$ in a certain sense ``isomorphic'' to $A$ having the property that (i) the diameter $X'$ of $A'$ is smaller than $X$, and (ii) the set $A'$ preserves the salient features of the solution set of the system of equations $P_1(\mathbf x)=\ldots =P_r(\mathbf x)=0$. We will report on our speculative meditations (both results and non-results) concerning this problem closely associated with the topic of Freiman homomorphisms.

number theory

Audience: researchers in the topic

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.

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