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 $O(n^{3/2})$.

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.