Skip to main content
Log in

On the symmetry measure of pseudorandom subsets

  • Published:
Periodica Mathematica Hungarica Aims and scope Submit manuscript

Abstract

Dartyge and Sárközy (partly with other coauthors) introduced pseudorandom measures of subsets. In this paper, we further study the symmetry measure of subsets by employing Gyarmati’s method. In addition, we study the symmetry measure of some subsets constructed by using power residues, additive characters and primitive roots.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Similar content being viewed by others

References

  1. J. Cassaigne, C. Mauduit, A. Sárközy, On finite pseudorandom binary sequences, VII: the measures of pseudorandomness. Acta Arith. 103, 97–118 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  2. Z. Chen, Large families of pseudo-random subsets formed by generalized cyclotomic classes. Monatsh. Math. 161, 161–172 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  3. C. Dartyge, E. Mosaki, A. Sárközy, On large families of subsets of the set of the integers not exceeding \(N\). Ramanujan J. 18, 209–229 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  4. C. Dartyge, A. Sárközy, On pseudo-random subsets of the set of the integers not exceeding \(N\). Period. Math. Hungar. 54, 183–200 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  5. C. Dartyge, A. Sárközy, Large families of pseudorandom subsets formed by power residues. Unif. Distrib. Theory 2, 73–88 (2007)

    MathSciNet  MATH  Google Scholar 

  6. C. Dartyge, A. Sárközy, M. Szalay, On the pseudo-randomness of subsets related to primitive roots. Combinatorica 30, 139–162 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  7. L. Goubin, C. Mauduit, A. Sárközy, Construction of large families of pseudorandom binary sequences. J. Number Theory 106, 56–69 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  8. K. Gyarmati, On a pseudorandom property of binary sequences. Ramanujan J. 8, 289–302 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  9. K. Gyarmati, On a family of pseudorandom binary sequences. Period. Math. Hungar. 49, 45–63 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  10. H. Liu, J. Gao, Large families of pseudorandom binary sequence constructed by using the Legendre symbol. Acta Arith. 154, 103–108 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  11. H. Liu, E. Song, A note on pseudorandom subsets formed by generalized cyclotomic classes. Publ. Math. Debr. 85, 257–271 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  12. C. Mauduit, C. Rivat, A. Sárközy, Construction of pseudorandom binary sequences using additive characters. Monatsh. Math. 141, 197–208 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  13. C. Mauduit, A. Sárközy, On finite pseudorandom binary sequence, I. Measure of pseudorandomness, the Legendre symbol. Acta Arith. 82, 365–377 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  14. L. Mérai, Construction of large families of pseudorandom binary sequences. Ramanujan J. 18, 341–349 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  15. H. Niederreiter, Statistical independence of nonlinear congruential pseudorandom numbers. Monatsh. Math. 106, 149–159 (1988)

    Article  MathSciNet  MATH  Google Scholar 

  16. W.M. Schmidt, Equations Over Finite Fields. An Elementary Approach, Lecture Notes in Mathematics, vol. 536 (Springer, New York, 1976)

    Book  Google Scholar 

  17. B. Sziklai, On the symmetry of finite pseudorandom binary sequences. Unif. Distrib. Theory 6, 143–156 (2011)

    MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Huaning Liu.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

This work is supported by the National Natural Science Foundation of China under Grant No. 12071368, and the Science and Technology Program of Shaanxi Province of China under Grants No. 2019JM-573 and 2020JM-026.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Jing, M., Liu, H. On the symmetry measure of pseudorandom subsets. Period Math Hung 86, 76–107 (2023). https://doi.org/10.1007/s10998-022-00461-x

Download citation

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10998-022-00461-x

Keywords

Mathematics Subject Classification

Navigation