Abstract
An upper bound for the multiplicative energy of the spectrum of an arbitrary subset of \(\mathbb{F}_{p}\) is obtained. Apparently, at present, this is the best bound.
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References
T. Tao and V. Vu, Additive Combinatorics (Cambridge Univ. Press, Cambridge, 2006).
T. F. Bloom, “A quantitative improvement for Roth’s theorem on arithmetic progressions,” J. Lond. Math. Soc. (2) 93 (3), 643–663 (2016).
M.-C. Chang, “A polynomial bound in Freiman’s theorem,” Duke Math. J. 113 (3), 399–419 (2002).
B. Green, “Some constructions in the inverse spectral theory of cyclic groups,” Combin. Probab. Comput. 12 (2), 127–138 (2003).
B. Green, “Spectral structure of sets of integers,” in Fourier Analysis and Convexity (Birkhäuser Boston, Boston, MA, 2004), pp. 83–96.
T. Sanders, “On certain other sets of integers,” J. Anal. Math. 116 (1), 53–82 (2012).
T. Sanders, “On Roth’s theorem on progressions,” Ann. of Math. (2) 174 (No. 1), 619–636 (2011).
I. D. Shkredov, “On sets of large exponential sums,” Izv. Ross. Akad. Nauk Ser. Mat. 72 (1), 161–182 (2008) [Izv. Math. 72 (1), 149–168 (2008)].
I. D. Shkredov, “On sumsets of dissociated sets,” Online J. Anal. Comb. 4, 1–26 (2009).
I. D. Shkredov, “An application of the sum-product phenomenon to sets avoiding several linear equations,” Mat. Sb. 209 (4), 117–142 (2018) [Sb. Math. 209 (4), 580–603 (2018)].
M. Rudnev, “On the number of incidences between planes and points in three dimensions,” Combinatorica 38 (1), 219–254 (2018).
M. Rudnev, I. D. Shkredov, and S. Stevens, On the Energy Variant of the Sum-Product Conjecture, arXiv:math.CO/1607.05053v2 (2016).
I. D. Shkredov, “On asymptotic formulae in some sum-product questions,” Mosc. J. Comb. Number Theory 8 (1), 15–41 (2019), arXiv:math.CO/1802.09066v2
P. Thang, M. Tait, and C. Timmons, “A Szemerédi-Trotter type theorem, sum-product estimates in finite quasifields and related results,” J. Combin. Theory Ser. A 147, 55–74 (2017).
L. A. Vinh, “A Szemerédi-Trotter type theorem and sum-product estimate over finite fields,” European J. Combin. 32 (8), 1177–1181 (2011).
B. Murphy, G. Petridis, Ol. Roche-Newton, M. Rudnev, and I. D. Shkredov, New Results on Sum-Product Type Growth Over Fields, arXiv:math.CO/1702.01003v2 (2017).
S. Stevens and F. de Zeeuw, “An improved point-line incidence bound over arbitrary fields,” Bull. Lond. Math. Soc. 49 (5), 842–858 (2017).
G. Petridis, Products of Differences in Prime-Order Finite Fields, arXiv:math.CO/1602.02142 (2016).
S. V. Konyagin and I. Shparlinski, Character Sums with Exponential Functions and Their Applications (Cambridge Univ. Press, Cambridge, 1999).
E. Aksoy Yazici, B. Murphy, M. Rudnev, and I. Shkredov, “Growth estimates in positive characteristic via collisions,” Int. Math. Res. Not. IMRN 2017 (23), 7148–7189 (2017).
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Shkredov, I.D. A Short Remark on the Multiplicative Energy of the Spectrum. Math Notes 105, 449–457 (2019). https://doi.org/10.1134/S0001434619030155
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DOI: https://doi.org/10.1134/S0001434619030155