Research log — Extremal affine copies of {0,1,3}

2026-08-17

• Ben Green, 100 Open Problems, Problem 24 (Dec 2025 update). SHA-256 e06971245914947f152550dee59bbb29fe0e798f0c51b2bc2557f824c2f9a44a. Statement: maximum number of affine translates of {0,1,3} in an n-element integer set. Comments: Ganguly / Pemantle; conjectured (1/3+o(1))n²; “Aaronson’s paper [2] should be consulted.” No later update on this problem in the Dec 2025 file.
• Green’s bibliography [2] is James Aaronson, Maximising the number of solutions to a linear equation in a set of integers, arXiv:1801.07135v4 = Bull. Lond. Math. Soc. 51 (2019), 577–594. The list entry 1805.01980 is a different paper (inverse scattering). SHA-256 of v4: f0da178874c50f7653d87271d86fae036abb5d19e9b83b17503251424f2e2ef2.
• Aaronson’s T(S) counts ordered triples in S³ with x+2y=3z, including n trivials. Affine copies of {0,1,3} are the nontrivial triples {y, y+d, y+3d}, d≠0. Asymptotic constant unaffected by the n trivials.
• Published record on γ_{1,2,-3} := limsup T(S)/|S|² after the fetch:
  • lower 1/3, from any interval (T = n²/3+O(1));
  • upper 3/4, from Hardy–Littlewood / Gabriel / Lev rearrangement (Aaronson (1.2) and Lemma 2.6). Exact γ known only for |abc|≤2. Aaronson conjectures γ_{1,2,-3}=1/3. No later paper found tonight moves this constant. Green Dec 2025 still lists the problem as open.
• Green–Sisask, arXiv:0709.4432, Theorem 1.2: maximum number of 3APs in an n-element integer set is ⌈n²/2⌉, by an ordering argument on midpoints. The same sum ∑(1+2 min(j−1,n−j)) appears in tonight’s {0,1,3} bound; the fibre injection is different (scale −2 about z, not reflection about z).
• Lev, On the number of solutions of a linear equation over finite sets, J. Combin. Theory Ser. A 83 (1998), 251–267; Lev–Pinchasi, Acta Arith. 163 (2014), 127–140: exact maximisers for x+y+z=0 and a±b=2c. Not used in the 1/2 argument.
• Bukh, Sums of dilates, Combin. Probab. Comput. 17 (2008): |S+2S|≥3|S|−o(|S|). Used only as residue (too weak alone to beat 3/4).

Bound claimed tonight

T(S) ≤ ⌈n²/2⌉ for every n-element S⊂ℤ, hence γ_{1,2,-3} ≤ 1/2. This is strictly below the published 3/4. It does not prove the conjecture 1/3. Independent check: compute/verify_half.py.

2026-08-27

• Green, 100 Open Problems, Problem 24 refetched. Dec 2025 text unchanged. SHA-256 still e06971245914947f152550dee59bbb29fe0e798f0c51b2bc2557f824c2f9a44a. No later update on this problem.
• Aaronson, arXiv:1801.07135v4 and the ar5iv HTML. Lemma 2.3 is T(S1,S2,S3)² ≤ E(a1 S1, a2 S2) |S3|; Lemma 2.6 is the 3/4 rearrangement. Still conjectures γ_{1,2,−3}=1/3. The energy comparison T² ≤ n E(S,2S) cannot beat 1/2 even if the interval maximises E: E_I ∼ 5n³/12 gives √(n E_I)/n² = √(5/12) ≈ 0.645.
• Green–Sisask, arXiv:0709.4432 HTML. Theorem 1.2 and the induction remark (endpoint 3AP degree). E(k,m), F(k,m) are the 3AP maximisers; they do not beat 1/3 for x+2y=3z.
• formal-conjectures Green #24 and PR #1772. Still records published γ ≤ 3/4; conjecture γ=1/3 open. Does not mention the 1/2 note in this repository.
• Aaronson thesis, ORA Chapter 3 is the same {0,1,3} discussion as the paper. No later numerical improvement.
• Failed lookup: arXiv / web search for a 2020–2026 paper that moves γ_{1,2,−3}. Nothing found. Forum leads not chased to a new paper.

q1 residue: endpoint 2/3-budget fails; no family with limsup T/n² > 1/3. Constant still 1/2. Cert: compute/q1/certs/q1.json.