Dense Sets with Prescribed Difference Representation Functions

Source abstract

For a set $A$ of positive integers, let $d_A(n)$ denote the number of representations of $n$ as a difference of two elements of $A$. We prove that, if $\liminf_{n\to\infty}f(n)\geq2$, then for every Sidon set $B$ and every positive function $ω(x)$ tending to infinity there is a set $A\subseteq\N$ such that $d_A(n)=f(n)$ for all $n\in\N$ and $|A(x)-B(x/2)|\leqω(x)$ for all $x\geq1$. We also give two extensions that allow $f(n)=1$ for infinitely many $n$. For an infinite Sidon set $B$, the same density scale $B(x/2)$ is retained when $f$ is at least $2$ on arbitrarily long intervals centered at elements of $2B$. If $\{n:f(n)\geq2\}$ contains arbitrarily long intervals, the density scale $B(x/3)$ is obtained for every Sidon set $B$. The proofs use remote insertion of new representations together with controlled deletions from a Sidon core. As a quantitative consequence, Ruzsa's dense infinite Sidon sequence gives $A(x)=x^{\sqrt2-1+o(1)}$ whenever $\{n:f(n)\geq2\}$ contains arbitrarily long intervals.