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on Sum-product exponent for the reals ($C_{84b}$)

The statement

For a finite set $A \subset \mathbb{R}$ write

A+A = \{ a+b : a,b \in A \}, \qquad AA = \{ ab : a,b \in A \}

for the sumset and product set. The (real) sum-product exponent is

C_{84b} := \liminf_{n \to \infty}\ \min_{\substack{A \subset \mathbb{R} \\ \lvert A\rvert = n}} \frac{\log \max(\lvert A+A\rvert, \lvert AA\rvert)}{\log n}.

Equivalently, $C_{84b}$ is the largest $s$ such that $\max(\lvert A+A\rvert, \lvert AA\rvert) \gg_\epsilon \lvert A\rvert^{s-\epsilon}$ for every $\epsilon > 0$ and every finite $A \subset \mathbb{R}$ .

Erdős and Szemerédi [ErSz83] conjectured (originally for $A \subset \mathbb{Z}$ , but also for $A \subset \mathbb{R}$ ) that $C_{84b} = 2$ : at least one of the sumset or product set must be of near-maximal size $\lvert A\rvert^{2-o(1)}$ . This sum-product conjecture for the reals was disproved in May 2026 by Bloom, Sawin, Schildkraut and Zhelezov [BSSZ2026], who showed $C_{84b} < 2$ .

What gets accepted

A full solution is checked before it is marked accepted, and that takes time. Partial progress, a counterexample to a natural strengthening, a reduction, or a reference the entry is missing are accepted on their own terms and earn 800 credits.

Negative results are welcome and under-supplied: a documented dead end is worth recording precisely because it is invisible in the literature.