Sum-product exponent for the reals

Upper bound, last-listed row

| $1.999281$ | [Al26], [EPF52] | Althoefer (28 May 2026), posted on the Erdős Problems forum [EPF52]; a ChatGPT 5.5 long-thinking optimization of the explicit constant of [BSSZ2026, §5] giving $c \ge 0.000719$. Unverified. |tracked since 2026-07-24 — no movement seen yet

Lower bound, last-listed row

| $\tfrac{4}{3} + \tfrac{10}{4407} \approx 1.335602$ | [Cu25] | Cushman; current record. |tracked since 2026-07-24 — no movement seen yet

Checked against material for its cited source

• Another row, not one shown above: bound row $2 - c$ for an absolute $c>0$ line 18 citing [BSSZ2026] — the claim we checked: that [BSSZ2026] constructs arbitrarily large sets A of real numbers with max(|A+A|, |AA|) at most |A|^(2-c) for an absolute c > 0, giving the upper bound 2-c listed in this row, and that a non-optimized explicit version gives c at least 0.00000087. The row’s description of the number-field construction is not checked here. What we read supports it: Theorem 1.1 gives arbitrarily large finite sets of real numbers with max(|A+A|, |AA|) at most |A| raised to the power 2 − c for an absolute c > 0, and the paper says a version of the proof with explicit constants, not optimized, gives c ≥ 0.00000087. source link (read in the arXiv preprint, version 1 (27 May 2026): the abstract and Theorem 1.1 on pp. 1–2 and the explicit constant on p. 12 from page images, and Section 5 on p. 11 as extracted text). Selected: drawn by its position in the list of cited rows, not because it looked wrong.

1 bound row(s) here have been checked against material for their cited source (1 drawn by position, 0 chosen because something already looked wrong). Across this whole ledger 70 row(s) have been drawn by position, against 691 rows that name a source, counted on 2026-09-27: 64 were checked against material for their cited sources and for 6 the cited source could not be read at all. 5 more were chosen for suspicion and checked; they are counted apart, because a set selected for suspicion carries no rate. A row not listed here has NOT been checked.