{"title":"Polynomial p-adic low-discrepancy sequences","authors":"Christian Weiß","doi":"10.1016/j.ffa.2025.102607","DOIUrl":null,"url":null,"abstract":"<div><div>The classic example of a low-discrepancy sequence in <span><math><msub><mrow><mi>Z</mi></mrow><mrow><mi>p</mi></mrow></msub></math></span> is <span><math><mo>(</mo><msub><mrow><mi>x</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>)</mo><mo>=</mo><mi>a</mi><mi>n</mi><mo>+</mo><mi>b</mi></math></span> with <span><math><mi>a</mi><mo>∈</mo><msubsup><mrow><mi>Z</mi></mrow><mrow><mi>p</mi></mrow><mrow><mo>×</mo></mrow></msubsup></math></span> and <span><math><mi>b</mi><mo>∈</mo><msub><mrow><mi>Z</mi></mrow><mrow><mi>p</mi></mrow></msub></math></span>. Here we address the non-linear case and show that a polynomial <em>f</em> generates a low-discrepancy sequence in <span><math><msub><mrow><mi>Z</mi></mrow><mrow><mi>p</mi></mrow></msub></math></span> if and only if it is a permutation polynomial <span><math><mspace></mspace><mrow><mi>mod</mi></mrow><mspace></mspace><mspace></mspace><msup><mrow><mi>p</mi></mrow><mrow><mn>2</mn></mrow></msup></math></span>. By this it is possible to construct non-linear examples of low-discrepancy sequences in <span><math><msub><mrow><mi>Z</mi></mrow><mrow><mi>p</mi></mrow></msub></math></span> for all primes <em>p</em>. Moreover, we prove a criterion which decides for any given polynomial in <span><math><msub><mrow><mi>Z</mi></mrow><mrow><mi>p</mi></mrow></msub></math></span> with <span><math><mi>p</mi><mo>∈</mo><mrow><mo>{</mo><mn>3</mn><mo>,</mo><mn>5</mn><mo>,</mo><mn>7</mn><mo>}</mo></mrow></math></span> if it generates a low-discrepancy sequence. We also discuss connections to the theories of Poissonian pair correlations and real discrepancy.</div></div>","PeriodicalId":50446,"journal":{"name":"Finite Fields and Their Applications","volume":"105 ","pages":"Article 102607"},"PeriodicalIF":1.2000,"publicationDate":"2025-03-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Finite Fields and Their Applications","FirstCategoryId":"100","ListUrlMain":"https://www.sciencedirect.com/science/article/pii/S1071579725000371","RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"MATHEMATICS","Score":null,"Total":0}
引用次数: 0
Abstract
The classic example of a low-discrepancy sequence in is with and . Here we address the non-linear case and show that a polynomial f generates a low-discrepancy sequence in if and only if it is a permutation polynomial . By this it is possible to construct non-linear examples of low-discrepancy sequences in for all primes p. Moreover, we prove a criterion which decides for any given polynomial in with if it generates a low-discrepancy sequence. We also discuss connections to the theories of Poissonian pair correlations and real discrepancy.
期刊介绍:
Finite Fields and Their Applications is a peer-reviewed technical journal publishing papers in finite field theory as well as in applications of finite fields. As a result of applications in a wide variety of areas, finite fields are increasingly important in several areas of mathematics, including linear and abstract algebra, number theory and algebraic geometry, as well as in computer science, statistics, information theory, and engineering.
For cohesion, and because so many applications rely on various theoretical properties of finite fields, it is essential that there be a core of high-quality papers on theoretical aspects. In addition, since much of the vitality of the area comes from computational problems, the journal publishes papers on computational aspects of finite fields as well as on algorithms and complexity of finite field-related methods.
The journal also publishes papers in various applications including, but not limited to, algebraic coding theory, cryptology, combinatorial design theory, pseudorandom number generation, and linear recurring sequences. There are other areas of application to be included, but the important point is that finite fields play a nontrivial role in the theory, application, or algorithm.