Condensation and densification for sets of large diameter

Abstract

Consider a set of integers $\mathscr A$ having finite diameter $X$, and a system of simultaneous polynomial equations to be solved over $\mathscr A$. In many circumstances, it is known that the number of solutions of this system is $O(X^\epsilon |\mathscr A|^\theta)$ for a suitable $\theta>0$ and any $\epsilon>0$. These estimates become worse than trivial when the diameter $X$ is very large compared to $|\mathscr A|$, or equivalently, when the set $\mathscr A$ is very sparse. This motivates the problem of seeking a new set of integers $\mathscr B$, in a certain sense isomorphic to $\mathscr A$, having the property that the diameter $X'$ of $\mathscr B$ is smaller than $X$, and at the same time the set $\mathscr B$ preserves the salient features of the solution set of the system of equations in question. We report on our speculative investigations concerning this problem closely associated with the topic of Freiman homomorphisms.