Abstract
By constructing suitable nonnegative exponential sums, we give upper bounds on the cardinality of any set \(B_q\) in cyclic groups \(\mathbb Z_q\) such that the difference set \(B_q-B_q\) avoids cubic residues modulo \(q\).
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References
S. D. Cohen, “Clique numbers of Paley graphs,” Quaest. Math. 11 (2), 225–231 (1988).
M. R. Gabdullin, “Sets in \(\mathbb Z_m\) whose difference sets avoid squares,” Sb. Math. 209 (11), 1603–1610 (2018) [transl. from Mat. Sb. 209 (11), 60–68 (2018)].
M. Matolcsi and I. Z. Ruzsa, “Difference sets and positive exponential sums. I: General properties,” J. Fourier Anal. Appl. 20 (1), 17–41 (2014).
R. C. Vaughan, The Hardy–Littlewood Method, 2nd ed. (Cambridge Univ. Press, Cambridge, 1997), Cambridge Tracts Math. 125.
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The authors are grateful to the reviewer for his/her suggestions which have improved the presentation of the paper.
Funding
M. Matolcsi was supported by the NKFIH grant nos. K132097 and K129335. I. Z. Ruzsa was supported by the NKFIH grant no. K129335.
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Published in Russian in Trudy Matematicheskogo Instituta imeni V.A. Steklova, 2021, Vol. 314, pp. 145–151 https://doi.org/10.4213/tm4190.
Dedicated to the memory of I. M. Vinogradov
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Matolcsi, M., Ruzsa, I.Z. Difference Sets and Positive Exponential Sums. II: Cubic Residues in Cyclic Groups. Proc. Steklov Inst. Math. 314, 138–143 (2021). https://doi.org/10.1134/S0081543821040088
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DOI: https://doi.org/10.1134/S0081543821040088