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.
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References
J. Cassaigne, C. Mauduit, A. Sárközy, On finite pseudorandom binary sequences, VII: the measures of pseudorandomness. Acta Arith. 103, 97–118 (2002)
Z. Chen, Large families of pseudo-random subsets formed by generalized cyclotomic classes. Monatsh. Math. 161, 161–172 (2010)
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)
C. Dartyge, A. Sárközy, On pseudo-random subsets of the set of the integers not exceeding \(N\). Period. Math. Hungar. 54, 183–200 (2007)
C. Dartyge, A. Sárközy, Large families of pseudorandom subsets formed by power residues. Unif. Distrib. Theory 2, 73–88 (2007)
C. Dartyge, A. Sárközy, M. Szalay, On the pseudo-randomness of subsets related to primitive roots. Combinatorica 30, 139–162 (2010)
L. Goubin, C. Mauduit, A. Sárközy, Construction of large families of pseudorandom binary sequences. J. Number Theory 106, 56–69 (2004)
K. Gyarmati, On a pseudorandom property of binary sequences. Ramanujan J. 8, 289–302 (2004)
K. Gyarmati, On a family of pseudorandom binary sequences. Period. Math. Hungar. 49, 45–63 (2004)
H. Liu, J. Gao, Large families of pseudorandom binary sequence constructed by using the Legendre symbol. Acta Arith. 154, 103–108 (2012)
H. Liu, E. Song, A note on pseudorandom subsets formed by generalized cyclotomic classes. Publ. Math. Debr. 85, 257–271 (2014)
C. Mauduit, C. Rivat, A. Sárközy, Construction of pseudorandom binary sequences using additive characters. Monatsh. Math. 141, 197–208 (2004)
C. Mauduit, A. Sárközy, On finite pseudorandom binary sequence, I. Measure of pseudorandomness, the Legendre symbol. Acta Arith. 82, 365–377 (1997)
L. Mérai, Construction of large families of pseudorandom binary sequences. Ramanujan J. 18, 341–349 (2009)
H. Niederreiter, Statistical independence of nonlinear congruential pseudorandom numbers. Monatsh. Math. 106, 149–159 (1988)
W.M. Schmidt, Equations Over Finite Fields. An Elementary Approach, Lecture Notes in Mathematics, vol. 536 (Springer, New York, 1976)
B. Sziklai, On the symmetry of finite pseudorandom binary sequences. Unif. Distrib. Theory 6, 143–156 (2011)
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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.
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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
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DOI: https://doi.org/10.1007/s10998-022-00461-x