×

On the squares in the set of elements of a finite field with constraints on the coefficients of its basis expansion. (English. Russian original) Zbl 1428.11046

Math. Notes 101, No. 2, 234-249 (2017); translation from Mat. Zametki 101, No. 2, 807-824 (2017).
Summary: Recent results of S. Dartyge, C. Mauduit, and A. Sárközy [Funct. Approximatio, Comment. Math. 52, No. 1, 65–74 (2015; Zbl 1321.11012)] concerning the problem of the number of squares among the elements of a finite field with constraints on the coefficients of its basis expansion are strengthened.

MSC:

11B75 Other combinatorial number theory
11D88 \(p\)-adic and power series fields
11T30 Structure theory for finite fields and commutative rings (number-theoretic aspects)

Citations:

Zbl 1321.11012
PDF BibTeX XML Cite
Full Text: DOI

References:

[1] Dartyge, C.; Mauduit, C.; Sárközy, A., Polynomial values and generators with missing digits in finite fields, Funct. Approx. Comment. Math., 52, 65-74, (2015) · Zbl 1321.11012
[2] Dartyge, C.; Sárközy, A., The sum of digits function in the finite field., Proc. Amer. Math. Soc., 141, 4119-4124, (2013) · Zbl 1354.11010
[3] Konyagin, S. V., Estimates of character sums in finite fields, Mat. Zametki, 88, 529-542, (2010) · Zbl 1261.11085
[4] R. Dietmann, C. Elsholtz, and I. E. Shparlinski, Prescribing the Binary Digits of Squarefree Numbers and Quadratic Residues, arXiv: 1601. 04754. · Zbl 1431.11015
[5] Gabdullin, M. R., On squares in special sets in finite fields, Chebyshevskii Sb., 17, 56-63, (2016)
[6] Wan, D., Generators and irreducible polynomials over finite fields, Math. Comp., 66, 1195-1212, (1997) · Zbl 0879.11072
[7] Winterhof, A., Character sums, primitive elements, and powers in finite fields, J. Number Theory, 91, 153-163, (2001) · Zbl 1008.11069
[8] Johnsen, J., On the distribution powers in finite fields, J. Reine Angew. Math., 251, 10-19, (1971) · Zbl 0237.10026
[9] Blackburn, S. R.; Konyagin, S. V.; Shparlinski, I. E., Counting additive decompositions of quadratic residues in finite fields, Funct. Approx. Comment. Math., 52, 223-227, (2015) · Zbl 1393.11006
[10] Tao, T.; Vu, V., Additive combinatorics, (2006), Cambridge · Zbl 1127.11002
[11] Betke, U.; Henk, M.; Wills, J. M., Successive-minima-type inequalities, Discrete Comput. Geom., 9, 165-175, (1993) · Zbl 0771.52007
[12] Banaszczyk, W., Inequalities for convex bodies and polar reciprocal lattices in R\^{}{n}, Discrete Comput. Geom., 13, 217-231, (1995) · Zbl 0824.52011
[13] Burgess, D. A., On character sums and primitive roots, Proc. Lond. Math. Soc. (3), 12, 179-192, (1962) · Zbl 0106.04003
[14] Karatsuba, A. A., On estimates of character sums, Izv. Akad. Nauk SSSR Ser. Mat., 34, 20-30, (1970)
[15] Chang, M.-Ch., Burgess inequality in \({F_{{p^2}}}\), Geom. Funct. Anal., 19, 1001-1016, (2009) · Zbl 1207.11083
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.