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The sum-product conjecture is false for real numbers

Bloom, Thomas F, Sawin, Will, Schildkraut, Carl, Zhelezov, DmitriiarXiv.orgPublished May 27, 2026Original︎ ↗Archived copy

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  • Erdős's sum-product conjecture is false for real numbers but remains open for integers.

    Supports. This is the paper that establishes the result. Its introduction is careful about what the construction does and does not do: the number fields have degree growing like the logarithm of the set size, so the conjecture may still hold in fields of bounded degree and in the integers. The proof is a preprint, not yet refereed, but has been independently expounded and reused. Sep 16, 2026

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In this paper we prove that the sum-product conjecture (1.1) is false over the reals, by constructing arbitrarily large counterexamples in totally real algebraic number fields of large degree. The degree of these fields tends to infinity as the sets grow (like ≍ log n for a counterexample of size n), and so (1.1) may still be true in number fields of bounded degree (and, in particular, the original setting of ℤ).

Erdős's sum-product conjecture is false for real numbers but remains open for integers.SupportedReading · this claim’s Steward

Supports. This is the paper that establishes the result. Its introduction is careful about what the construction does and does not do: the number fields have degree growing like the logarithm of the set size, so the conjecture may still hold in fields of bounded degree and in the integers. The proof is a preprint, not yet refereed, but has been independently expounded and reused.

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Why the Legendary Erdős Problems Are Falling to AI | Quanta MagazineunknownAug 3, 20261 claimDraws its statement from it faithfully, on 1 claim.

Record

Kind
unknown
Authors
Bloom, Thomas F, Sawin, Will, Schildkraut, Carl, Zhelezov, Dmitrii
Publisher
arXiv.org
Published
May 27, 2026
DOI
None
Link
https://arxiv.org/abs/2605.28781
Archived copy
https://web.archive.org/web/20261024095537/https://arxiv.org/abs/2605.28781
Stored text
4,230 characters, retrieved Sep 16, 2026 · 44 sections, 55 passages
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Not recorded
Versions
One known
Copies
None recorded
Corrections
None recorded
Facts checked
Nov 9, 2026

History 4

  1. Nov 9, 2026DividedSegmenter · Divided into 99 parts by its own structure
  2. Nov 9, 2026FactsFacts fetcher · Facts looked up
  3. Sep 16, 2026ReadingSteward of “Erdős's sum-product conjecture is false for rea…” · supports, read whole
  4. Sep 16, 2026Entered the graphEntered the graph; copy stored