The distribution of απ modulo one in Fq(T): beyond Matomäki's exponent

Abstract

Let α be an element of Fq((1/T)) which is not in Fq(T). We ask how small nrmαπ can be made when π runs over irreducible polynomials, and write θ(α) for the largest exponent θ such that nrmαπ<|π|^−θ has infinitely many solutions. The analogue of Matomäki's exponent, θ(α)≥1/3, was proved by Baier and the author, and it is the natural limit of the Riemann Hypothesis. In this paper we go beyond it. For odd characteristic p and q large in terms of p and θ, we prove that θ(α)≥θ for every α and every θ<√2−1=0.414…, and for every θ<1/2 when α has bounded partial quotients. The main tools are an exact count of points in affine subspaces of Fq[T], a splitting of these subspaces into long arithmetic progressions, and the results of Sawin and Shusterman on the Möbius function along such progressions.

Linear-exponent bounds for the automatic complexity of rational diagonals modulo prime powers
Alex Shvets
arXiv1 Nov 2026PreprintCC BY 4.0Full textGeneral mathematicsarXiv
On the distribution of αp³ modulo one over special primes
Stoyan Dimitrov, Meglena Lazarova
HAL24 Sep 2026PreprintCC BY 4.0Full textGeneral mathematicsHAL
The distribution of αp modulo one in real quadratic function fields
Esrafil Ali Molla
arXiv16 Oct 2026PreprintarXiv licenceFull textAlgebra & number theoryarXiv
Growth beyond exponent 3/2 for convexity and iterated sum sets
Michalis Kokkinos, Oliver Roche-Newton
arXiv14 Sep 2026PreprintCC BY 4.0Full textAlgebra & number theory Combinatorics & discrete mathematicsarXiv
The Hénon equation beyond the critical Sobolev exponent
Francesca Gladiali, Massimo Grossi
arXiv29 Oct 2026PreprintarXiv licenceFull textAnalysis & PDEsarXiv

Cite

BibTeX RIS

BibTeX

@misc{molla2026distribution,
  title = {{The distribution of $απ$ modulo one in $\Fq(T)$: beyond Matomäki's exponent}},
  author = {Esrafil Ali Molla},
  howpublished = {arXiv},
  year = {2026},
  month = nov,
  doi = {10.48550/arxiv.2611.00623},
  eprint = {2611.00623},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2611.00623},
}

RIS

TY  - UNPB
TI  - The distribution of απ modulo one in Fq(T): beyond Matomäki's exponent
AU  - Esrafil Ali Molla
PY  - 2026
DA  - 2026/11/01
PB  - arXiv
DO  - 10.48550/arxiv.2611.00623
UR  - https://arxiv.org/abs/2611.00623
AB  - Let α be an element of Fq((1/T)) which is not in Fq(T). We ask how small nrmαπ can be made when π runs over irreducible polynomials, and write θ(α) for the largest exponent θ such that nrmαπ<|π|^−θ has infinitely many solutions. The analogue of Matomäki's exponent, θ(α)≥1/3, was proved by Baier and the author, and it is the natural limit of the Riemann Hypothesis. In this paper we go beyond it. For odd characteristic p and q large in terms of p and θ, we prove that θ(α)≥θ for every α and every θ<√2−1=0.414…, and for every θ<1/2 when α has bounded partial quotients. The main tools are an exact count of points in affine subspaces of Fq[T], a splitting of these subspaces into long arithmetic progressions, and the results of Sawin and Shusterman on the Möbius function along such progressions.
ER  -

Abstract

Related

Cite