Erdős Problem #1097

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.