The sum-product conjecture is false for real numbers

2026/05/27 by Thomas F Bloom, Will Sawin, Carl Schildkraut +1 · 7 voices · 4 citations
#math.NT #math.CO

paper · pdf

Abstract

We disprove the sum-product conjecture for real numbers by constructing arbitrarily large A⊂ ℝ (whose elements are algebraic integers in a number field of degree \asymp log| A|) such that max(| A+A| ,| AA|)≤ | A|^2-c where c>0 is an absolute constant. We also disprove the many sums and products conjecture by constructing, for any k≥ 3, arbitrarily large A⊂ ℝ such that max(| kA|,| A^(k)|)≤ | A|^{C(log k)/(loglog k)} for some constant C>0. We obtain similar constructions for p-adics, finite fields, and function fields in positive characteristic, and also obtain new lower bounds for the number of solutions to linear equations in a multiplicative group and the number of solutions to the unit equation in sufficiently many variables.

Citations

• Remarks on the disproof of the unit distance conjecture
• Linear Equations in Variables which Lie in a Multiplicative Group

Cited by

• The sharp exponent for the minimal distance problem
• Autonomous disproofs of the sum-product conjecture over \mathbb R with GPT-5.5 Pro
• On an asymmetric additive energy inequality
• Diameter bounds for finite simple Lie algebras
• Erdős–Szemerédi theorem [wikipedia]

Discussions

• “The sum-product conjecture is false for real numbers” THOMAS F. BLOOM, WILL SAWIN, CARL SCHILDKRAUT, AND DMITRII ZHELEZOV A human proof that exploits the same kind of “tower of fields” that was used [bsky, 45 points, 4 comments]
• Paper: [bsky, 21 points, 1 comments]
• The Sum-Product conjecture is false for real numbers [hn, 13 points, 0 comments]
• … Si on s'intéresse à la combinatoire ou aux problèmes d'Erdős, c'est évidemment un résultat important; mais depuis il y en a un autre d'importance comparable (le problème sum-product sur ℝ) qui a été [bsky, 3 points, 1 comments]
• Humans have disproved the sum-product conjectures for real numbers [hn, 3 points, 1 comments]
• Là, c'est une conjecture d'Erdos qui a été invalidée par des humain arxiv.org/abs/2605.28781 mais les auteurs indiquent, dans une partie "the role of AI", que le contre exemple produit par OpenAI sur [bsky, 1 points, 1 comments]
• Did you see the big paper that just got released? arxiv.org/abs/2605.28781 [bsky, 0 points, 0 comments]

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