Quadratic threshold for odd-sized local difference sets

Determine the exact quadratic threshold for the difference-set local properties function g(n,k,ell) when k is odd, including the asymptotic behavior of g(n,k,ell) for ell between (k+1)^2/4 - 4 and (k+1)^2/4.

Background

For the local difference-set problem, g(n,k,ell) denotes the minimum number of distinct positive differences in an n-element set of real numbers such that every k-element subset has at least ell distinct differences. The quadratic threshold is the smallest ell for which g(n,k,ell)=Omega(n2).

The paper recalls that for odd k, the previously known results establish g(n,k,(k+1)2/4)=Omega(n2) and g(n,k,(k+1)2/4-4)=o(n2). The paper improves the latter subquadratic bound to O(nc) for an absolute constant c<2, but does not determine the threshold or the behavior at the intermediate values of ell.

References

Unlike for even $k$, we do not know the exact quadratic threshold for odd $k$ --- we know that $g(n, k, (k + 1)2/4) = (n2)$, but we do not understand the behavior of $g(n, k, )$ when $$ is between these two values.

— Exponents in the local properties problem for difference sets have a gap at 2  (2501.11148 - Das, 19 Jan 2025) in Section 1, subsection “Our question and result,” paragraph following Proposition 1.2