Limit distribution of algebraic integral points on curves

Abstract

For a quasi-projective arithmetic surface 𝒰/ℤ and a compact subset E⊂ 𝒰(ℂ) under mild regularity assumptions, we study algebraic integral points on 𝒰 whose Galois orbits lie in E. We characterize the collections of local probability measures that arise simultaneously as the local limit distributions of Galois orbits of such points. Our result also works for measures prescribed at a subset of places. This generalizes the results of Smith and Orloski--Sardari on 𝒰=𝔸¹ concerning the archimedean place. As a consequence, we prove an integral version of Szachniewicz's theorem on curves, which states that every \emph{integral} GVF functional can be approximated by a sequence of algebraic integral points in the GVF topology. We also give several applications: we prove that the essential minimum of height functions on curves can be attained by algebraic integral points; we connect integer Chebyshev constants with the essential minima of certain height functions on 𝔸¹ and prove a conjecture of Montgomery; we also answer affirmatively a question of Levenberg--Londhe by showing that the smallest limit of averaged trace of totally positive algebraic integers can be attained by a sequence of totally positive algebraic units.

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BibTeX RIS

BibTeX

@misc{qu2026limit,
  title = {{Limit distribution of algebraic integral points on curves}},
  author = {Binggang Qu and Chengyuan Yang},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.39505},
  eprint = {2609.39505},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.39505},
  note = {CC BY 4.0},
}

RIS

TY  - UNPB
TI  - Limit distribution of algebraic integral points on curves
AU  - Binggang Qu
AU  - Chengyuan Yang
PY  - 2026
DA  - 2026/09/30
PB  - arXiv
DO  - 10.48550/arxiv.2609.39505
UR  - https://arxiv.org/abs/2609.39505
AB  - For a quasi-projective arithmetic surface 𝒰/ℤ and a compact subset E⊂ 𝒰(ℂ) under mild regularity assumptions, we study algebraic integral points on 𝒰 whose Galois orbits lie in E. We characterize the collections of local probability measures that arise simultaneously as the local limit distributions of Galois orbits of such points. Our result also works for measures prescribed at a subset of places. This generalizes the results of Smith and Orloski--Sardari on 𝒰=𝔸^1 concerning the archimedean place. As a consequence, we prove an integral version of Szachniewicz's theorem on curves, which states that every \emph{integral} GVF functional can be approximated by a sequence of algebraic integral points in the GVF topology. We also give several applications: we prove that the essential minimum of height functions on curves can be attained by algebraic integral points; we connect integer Chebyshev constants with the essential minima of certain height functions on 𝔸^1 and prove a conjecture of Montgomery; we also answer affirmatively a question of Levenberg--Londhe by showing that the smallest limit of averaged trace of totally positive algebraic integers can be attained by a sequence of totally positive algebraic units.
ER  -

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