## Additive energy and the metric Poissonian property

Research output: Contribution to journalArticle

### Standard

Additive energy and the metric Poissonian property. / Bloom, Thomas F.; Chow, Samuel Khai Ho; Gafni, Ayla; Walker, Aled.

In: Mathematika, Vol. 64, No. 3, 19.06.2018, p. 679-700.

Research output: Contribution to journalArticle

### Harvard

Bloom, TF, Chow, SKH, Gafni, A & Walker, A 2018, 'Additive energy and the metric Poissonian property', Mathematika, vol. 64, no. 3, pp. 679-700. https://doi.org/10.1112/S0025579318000207

### APA

Bloom, T. F., Chow, S. K. H., Gafni, A., & Walker, A. (2018). Additive energy and the metric Poissonian property. Mathematika, 64(3), 679-700. https://doi.org/10.1112/S0025579318000207

### Vancouver

Bloom TF, Chow SKH, Gafni A, Walker A. Additive energy and the metric Poissonian property. Mathematika. 2018 Jun 19;64(3):679-700. https://doi.org/10.1112/S0025579318000207

### Author

Bloom, Thomas F. ; Chow, Samuel Khai Ho ; Gafni, Ayla ; Walker, Aled. / Additive energy and the metric Poissonian property. In: Mathematika. 2018 ; Vol. 64, No. 3. pp. 679-700.

@article{77af0d51ce0a44b7a36a31d00e91de18,
title = "Additive energy and the metric Poissonian property",
abstract = "Let $A$ be a set of natural numbers. Recent work has suggested a strong linkbetween the additive energy of $A$ (the number of solutions to $a_1 + a_2 = a_3+ a_4$ with $a_i \in A$) and the metric Poissonian property, which is afine-scale equidistribution property for dilates of $A$ modulo $1$. Thereappears to be reasonable evidence to speculate a sharp Khintchine-typethreshold, that is, to speculate that the metric Poissonian property should becompletely determined by whether or not a certain sum of additive energies isconvergent or divergent. In this article, we primarily address the convergencetheory, in other words the extent to which having a low additive energy forcesa set to be metric Poissonian.",
keywords = "pair correlations, distribution modulo 1, metric diophantine approximation, additive combinatorics, large deviations",
author = "Bloom, {Thomas F.} and Chow, {Samuel Khai Ho} and Ayla Gafni and Aled Walker",
note = "{\circledC} 2018 University College London. This is an author-produced version of the published paper. Uploaded in accordance with the publisher’s self-archiving policy. Further copying may not be permitted; contact the publisher for details",
year = "2018",
month = "6",
day = "19",
doi = "10.1112/S0025579318000207",
language = "English",
volume = "64",
pages = "679--700",
journal = "Mathematika",
issn = "0025-5793",
publisher = "University College London",
number = "3",

}

TY - JOUR

T1 - Additive energy and the metric Poissonian property

AU - Bloom, Thomas F.

AU - Chow, Samuel Khai Ho

AU - Gafni, Ayla

AU - Walker, Aled

N1 - © 2018 University College London. This is an author-produced version of the published paper. Uploaded in accordance with the publisher’s self-archiving policy. Further copying may not be permitted; contact the publisher for details

PY - 2018/6/19

Y1 - 2018/6/19

N2 - Let $A$ be a set of natural numbers. Recent work has suggested a strong linkbetween the additive energy of $A$ (the number of solutions to $a_1 + a_2 = a_3+ a_4$ with $a_i \in A$) and the metric Poissonian property, which is afine-scale equidistribution property for dilates of $A$ modulo $1$. Thereappears to be reasonable evidence to speculate a sharp Khintchine-typethreshold, that is, to speculate that the metric Poissonian property should becompletely determined by whether or not a certain sum of additive energies isconvergent or divergent. In this article, we primarily address the convergencetheory, in other words the extent to which having a low additive energy forcesa set to be metric Poissonian.

AB - Let $A$ be a set of natural numbers. Recent work has suggested a strong linkbetween the additive energy of $A$ (the number of solutions to $a_1 + a_2 = a_3+ a_4$ with $a_i \in A$) and the metric Poissonian property, which is afine-scale equidistribution property for dilates of $A$ modulo $1$. Thereappears to be reasonable evidence to speculate a sharp Khintchine-typethreshold, that is, to speculate that the metric Poissonian property should becompletely determined by whether or not a certain sum of additive energies isconvergent or divergent. In this article, we primarily address the convergencetheory, in other words the extent to which having a low additive energy forcesa set to be metric Poissonian.

KW - pair correlations

KW - distribution modulo 1

KW - metric diophantine approximation

KW - large deviations

U2 - 10.1112/S0025579318000207

DO - 10.1112/S0025579318000207

M3 - Article

VL - 64

SP - 679

EP - 700

JO - Mathematika

JF - Mathematika

SN - 0025-5793

IS - 3

ER -