Given a finite set of integers A (modelled as a Finset ℤ), the set CommonDifferencesThreeTermAP A consists of all integers d such that there is a non-trivial three-term arithmetic progression a, b, c ∈ A with b - a = d and c - b = d.

The main conjecture: for any finite set of integers $A$ with $|A| = n$, the number of distinct common differences in three-term arithmetic progressions is $O(n^{3/2})$.

This conjecture was resolved negatively by showing that the problem is exactly equivalent to Bourgain's sums-differences question [Bo99], which was introduced as an arithmetic path towards the Kakeya conjecture. Under this equivalence:

The greatest achievable exponent for this problem is equal to the smallest constant $c$ achievable for Bourgain's sums-differences question: $$|A -_G B| \ll \max(|A|, |B|, |A +_G B|)^c$$

The $O(n^{3/2})$ prediction is disproved because the lower bound has been shown to satisfy $c \ge 1.77898$ (due to Zheng and AlphaEvolve [GGTW25], improving on Lemm [Le15]), which is strictly greater than $3/2 = 1.5$.

The best known upper bound is $c \le 11/6 \approx 1.833$ (due to Katz and Tao [KaTa99]).

While the specific $O(n^{3/2})$ prediction is resolved negatively, the general question of determining the exact optimal exponent $c$ remains open.

A weaker bound has been proven: there are always at most $n^2$ such values of $d$.

A trivial lower bound: for sufficiently large n there exist sets $A$ with $|A| = n$ that contain at least $\Omega(n)$ distinct common differences of three-term arithmetic progressions.