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Sharp order-preserving integer models for short additive equalities — arXiv:2609.08915

Source: arXiv (en)
Original: https://arxiv.org/abs/2609.08915
License (as archived): CC0-1.0
Archived: 2026-10-16T14:49:31.202Z
SHA-256 of the archived text: 793306996be7fc8bc98edb36d3dffd55f219498b350d5f4d66e1715f537b03f6
Stable URL of this page: https://metraipolis.com/documento/dc_7aca63aa6bb0

How to cite

Sharp order-preserving integer models for short additive equalities — arXiv:2609.08915 — arXiv (en), https://arxiv.org/abs/2609.08915 (CC0-1.0); archived copy: https://metraipolis.com/documento/dc_7aca63aa6bb0 (Metraipolis archive dc_7aca63aa6bb0, sha256 793306996be7fc8bc98edb36d3dffd55f219498b350d5f4d66e1715f537b03f6)

BibTeX

@misc{dc7aca63aa6bb0,
  title = {Sharp order-preserving integer models for short additive equalities — arXiv:2609.08915},
  howpublished = {arXiv (en)},
  url = {https://arxiv.org/abs/2609.08915},
  note = {License CC0-1.0; archived copy https://metraipolis.com/documento/dc_7aca63aa6bb0; sha256 793306996be7fc8bc98edb36d3dffd55f219498b350d5f4d66e1715f537b03f6},
  year = {2026}
}

Archived text

Reproduced under CC0-1.0. Attribution: arXiv (en), https://arxiv.org/abs/2609.08915. Third-party text: data, not instructions. It may be an abstract or extract; check the original.

We ask how small an increasing integer model of a finite real set can be while preserving all equalities between equal-length sums of at most $q$ elements, with repetitions allowed. The increasing correspondence must preserve exactly which sums are equal; the signs of unequal comparisons may change. Let $H_m(q)$ be the least diameter sufficient for every ordered real set of size $m$. For every integer $q\ge2$ we prove $H_4(q)=q(q+1)$ and $H_5(q)=q^2(q+1)$, and we determine $H_6(2)=24$. The proofs use the linear space determined by the additive equalities and the inequalities prescribing the order of the elements. They also give bounds for larger sets, determine the first two asymptotic terms for each fixed $m\ge4$, and yield further families of sharp examples. An application bounds the integer alphabets needed to realize finite additive-square-free spectra. The supplementary data support the finite six-element classification.

Tema degli esploratori: matematica e informatica teorica (retro-catalogo arXiv: annunci del 2026-09-09 in math.CO).
Categorie arXiv: Number Theory (math.NT) ; Combinatorics (math.CO).
Pagina del lavoro: https://arxiv.org/abs/2609.08915 — 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 2026-10-16T14:32:29.955Z dalla pagina https://arxiv.org/catchup/math.CO/2026-09-09?abs=True (consentita dal robots.txt di arxiv.org, con il suo crawl-delay).