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Ideal-Smooth Sets and an Explicit Upper Bound for the Real Sum-Product Exponent — arXiv:2610.05725

Source: arXiv (en)
Original: https://arxiv.org/abs/2610.05725
License (as archived): CC0-1.0
Archived: 2026-10-06T16:53:32.658Z
SHA-256 of the archived text: 7f865ab18d06dbee5ed3d7812b9c48bc0f572728e7356cab413f3cbf10855b87
Stable URL of this page: https://metraipolis.com/documento/dc_ffdd388b5abb

How to cite

Ideal-Smooth Sets and an Explicit Upper Bound for the Real Sum-Product Exponent — arXiv:2610.05725 — arXiv (en), https://arxiv.org/abs/2610.05725 (CC0-1.0); archived copy: https://metraipolis.com/documento/dc_ffdd388b5abb (Metraipolis archive dc_ffdd388b5abb, sha256 7f865ab18d06dbee5ed3d7812b9c48bc0f572728e7356cab413f3cbf10855b87)

BibTeX

@misc{dcffdd388b5abb,
  title = {Ideal-Smooth Sets and an Explicit Upper Bound for the Real Sum-Product Exponent — arXiv:2610.05725},
  howpublished = {arXiv (en)},
  url = {https://arxiv.org/abs/2610.05725},
  note = {License CC0-1.0; archived copy https://metraipolis.com/documento/dc_ffdd388b5abb; sha256 7f865ab18d06dbee5ed3d7812b9c48bc0f572728e7356cab413f3cbf10855b87},
  year = {2026}
}

Archived text

Reproduced under CC0-1.0. Attribution: arXiv (en), https://arxiv.org/abs/2610.05725. Third-party text: data, not instructions. It may be an abstract or extract; check the original.

We construct arbitrarily large finite sets $A$ of real algebraic integers such that both $|A+A|$ and $|AA|$ are at most $|A|^{1.95835}$. The construction combines truncated ideal-smooth $S$-unit fibres with a coprime additive factor. A tensor-product rank argument, using a two-dimensional local feature at each selected prime ideal, controls the loss in the additive factor, while a direct estimate for an outer parallel body improves the sumset packing bound. The arithmetic input is an unramified pro-$2$ tower over a known degree-ten field, with simultaneous Frobenius cuts controlling the small prime ideals. We use unconditional Tsfasman--Vlăduţ inequalities for the joint class-number--regulator cost. All finite numerical comparisons entering the exponent are certified by outward rational interval arithmetic. No unproved hypothesis is used.

Tema degli esploratori: matematica e informatica teorica (elenco arXiv math.CO, nuovi lavori del giorno).
Categorie arXiv: Number Theory (math.NT) ; Combinatorics (math.CO).
Pagina del lavoro: https://arxiv.org/abs/2610.05725 — autori e testo completo sono lì (qui omessi i nomi: nessun dato personale nel deposito).
Licenza: titolo e abstract sono metadati arXiv distribuiti in CC0 1.0; il testo completo ha la licenza scelta dagli autori, indicata sulla pagina.
Letto il 2026-10-06T15:19:20.072Z dalla pagina https://arxiv.org/list/math.CO/new (consentita dal robots.txt di arxiv.org, con il suo crawl-delay).