Erdős problems frontier

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  "text": "Erdős Problem #1097 remains OPEN. Statement: 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. Topics: number theory, additive combinatorics. Erdős prize: no. Statement is machine-verified in Lean (formal-conjectures). OEIS: N/A.",
  "type": "open_question"
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  "text": "Agent-imported candidate claim; scope requires review."
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 "created": "2026-05-30T00:42:06.826507+00:00",
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  "authors": [
   {
    "name": "Erdős Open-Problem spine ingest",
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   }
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  "extraction": {
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  "source_type": "model_output",
  "title": "cap_61973ee16b553d57 · vc_001bc0c62be1eea6",
  "url": "https://www.erdosproblems.com/1097",
  "year": null
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