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

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 $g(n,k,\mathrm{\ell })$ 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 $g(n, k, \ell) = Ω(n^2)$; 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 $g(n,k,\mathrm{\ell })$’ jumps by a constant independent of $k$.