Exponents in the local properties problem for difference sets have a gap at 2

TL;DR

This paper investigates the minimum number of differences in large sets of real numbers with local properties, revealing a gap in the exponent of the growth rate at a critical threshold for even subset sizes.

Contribution

It establishes that at the quadratic threshold for the local properties problem, the growth exponent of the difference count jumps by a constant less than 2, showing a gap in the exponent behavior.

Findings

For even k, when ll is just below the quadratic threshold, g(n, k, ll) = O(n^c) with c < 2.

The exponent of n in g(n, k, ll) jumps by a constant at the quadratic threshold.

The result demonstrates a gap in the asymptotic behavior of difference sets at a critical local property threshold.

Abstract

We study the local properties problem for difference sets: If we have a large set of real numbers and know that every small subset has many distinct differences, to what extent must the entire set have many distinct differences? More precisely, we define $g(n, k, \ell)$ to be the minimum number of differences in an $n$-element set with the `local property' that every $k$-element subset has at least $\ell$ differences; we study the asymptotic behavior of $g(n, k, \ell)$ as $k$ and $\ell$ are fixed and $n \to \infty$. The quadratic threshold is the smallest $\ell$ (as a function of $k$) for which $g(n, k, \ell) = \Omega(n^2)$; its value is known when $k$ is even. In this paper, we show that for $k$ even, when $\ell$ is one below the quadratic threshold, we have $g(n, k, \ell) = O(n^c)$ for an absolute constant $c < 2$ -- i.e., at the quadratic threshold, the `exponent of $n$ in $g(n, k, \ell)$' jumps by a constant independent of $k$.