Abstract
Dartyge and Sárközy (partly with other coauthors) introduced pseudorandom measures of subsets, and studied the pseudorandomness of certain subsets related to primitive roots. In this paper we answer a conjecture of Dartyge, Sárközy and Szalay, and study the pseudorandom properties of some subsets constructed by using primitive roots.
Similar content being viewed by others
References
Carlitz, L.: Sets of primitive roots. Compos. Math. 13, 65–70 (1956)
Chen, Z.: Large families of pseudo-random subsets formed by generalized cyclotomic classes. Monatshefte Math. 161, 161–172 (2010)
Dartyge, C., Sárközy, A.: On pseudo-random subsets of the set of the integers not exceeding \(N\). Period. Math. Hung. 54, 183–200 (2007)
Dartyge, C., Sárközy, A.: Large families of pseudorandom subsets formed by power residues. Unif. Distrib. Theory 2, 73–88 (2007)
Dartyge, C., Mosaki, E., Sárközy, A.: On large families of subsets of the set of the integers not exceeding \(N\). Ramanujan J. 18, 209–229 (2009)
Dartyge, C., Sárközy, A., Szalay, M.: On the pseudo-randomness of subsets related to primitive roots. Combinatorica 30, 139–162 (2010)
Gyarmati, K., Sárközy, A.: Equations in finite fields with restricted solution sets. I. Character sums. Acta Math. Hung. 118, 129–148 (2008)
Liu, H., Song, E.: A note on pseudorandom subsets formed by generalized cyclotomic classes. Publ. Math. Debr. 85, 257–271 (2014)
Mauduit, C., Sárközy, A.: On finite pseudorandom binary sequencs I: measure of pseudorandomness, the Legendre symbol. Acta Arith. 82, 365–377 (1997)
Mérai, L.: Construction of large families of pseudorandom binary sequences. Ramanujan J. 18, 341–349 (2009)
Schmidt, W.M.: Equations over Finite Fields. An Elementary Approach. Lecture Notes in Mathematics, vol. 536. Springer, New York (1976)
Szalay, M.: On the distribution of primitive roots mod \(p\). Mat. Lapok. 21, 357–362 (1970)
Ta, T.: On the distributions of quadratic residues and primitive roots over finite fields. Studia Sci. Math. Hung. 54, 426–435 (2017)
Tanti, J., Thangadurai, R.: Distribution of residues and primitive roots. Proc. Indian Acad. Sci. Math. Sci. 123, 203–211 (2013)
Vegh, E.: Pairs of consecutive primitive roots modulo a prime. Proc. Am. Math. Soc. 19, 1169–1170 (1968)
Acknowledgements
The authors express their gratitude to the referee for his/her helpful and detailed comments.
Author information
Authors and Affiliations
Corresponding author
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. 11571277, and the Science and Technology Program of Shaanxi Province of China under Grant No. 2019JM-573 and 2020JM-026.
Rights and permissions
About this article
Cite this article
Liu, H., Jing, M. On the pseudorandom properties of subsets constructed by using primitive roots. Ramanujan J 56, 533–553 (2021). https://doi.org/10.1007/s11139-020-00317-3
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11139-020-00317-3