Abstract
We improve an estimate for the additive energy of sets \(A\) with small product \(AA\). The proof uses some properties of level sets of convolutions of the indicator function of \(A\), namely, their almost invariance under multiplication by elements of \(A\).
Similar content being viewed by others
References
J. Bourgain, “More on the sum-product phenomenon in prime fields and its applications,” Int. J. Number Theory 1 (1), 1–32 (2005).
J. Bourgain, “Estimates on exponential sums related to the Diffie–Hellman distributions,” Geom. Funct. Anal. 15 (1), 1–34 (2005).
S. V. Konyagin, “Estimates for trigonometric sums over subgroups and for Gauss sums,” in IV International Conference “Modern Problems of Number Theory and its Applications”: Current Problems, Pt. III (Moskov. Gos. Univ., Moscow, 2002), pp. 86–114 [in Russian].
J. B. Friedlander, J. Hansen, and I. E. Shparlinski, “Character sums with exponential functions,” Mathematika 47 (1–2), 75–85 (2000).
J. B. Friedlander, J. Hansen, and I. E. Shparlinski, “Stronger sum-product inequalities for small sets,” Proc. Amer. Math. Soc. 148 (4), 1467–1479 (2020).
G. Shakan, On Higher Energy Decomposition and the Sum-Product Phenomenon, arXiv: 1803.04637 (2018).
J. Bourgain and M.-Ch. Chang, “On the size of \(k\)-fold sum and product sets of integers,” J. Amer. Math. Soc. 17 (2), 473–497 (2004).
I. D. Shkredov, “Some remarks on the asymmetric sum-product phenomenon,” Mosc. J. Comb. Number Theory 8 (1), 15–41 (2019).
B. Murphy, M. Rudnev, I. D. Shkredov, and Yu. N. Shteinikov, “On the few products, many sums problem,” J. Theor. Nombres Bordeaux 31 (3), 573–603 (2019).
I. D. Shkredov, “Some remarks on sets with small quotient set,” Sb. Math. 208 (12), 1854–1868 (2017).
T. Schoen and I. D. Shkredov, “Higher moments of convolutions,” J. Number Theory 133 (5), 1693–1737 (2013).
I. D. Shkredov, “Some new results on higher energies,” in Trans. Moscow Math. Soc. (2013), Vol. 74, pp. 31–63.
I. V. Vyugin and I. D. Shkredov, “On additive shifts of multiplicative subgroups,” Sb. Math. 203 (6), 844–863 (2012).
T. Tao and V. Vu, Additive Combinatorics, in Cambridge Stud. Adv. Math. (Cambridge Univ. Press, Cambridge, 2006), Vol. 105.
I. D. Shkredov, “Some remarks on the Balog–Wooley decomposition theorem and quantities\(D^{+}\), \(D^\times\),” Proc. Steklov Inst. Math. 298 (suppl. 1), 74–90 (2017).
E. Szemerédi and W. T. Trotter, “Extremal problems in discrete geometry,” Combinatorica 3 (3–4), 381–392 (1983).
I. D. Shkredov, “On sums of Szemerédi–Trotter sets,” Proc. Steklov Inst. Math. 289, 300–309 (2015).
G. Elekes, “On the number of sums and products,” Acta Arith. 81 (4), 365–367 (1997).
J. Bourgain, N. Katz, and T. Tao, “A sum-product estimate in finite fields, and applications,” Geom. Funct. Anal. 14 (1), 27–57 (2004).
Acknowledgments
The authors wish to express gratitude to T. Schoen for valuable remarks.
Funding
This work was supported by the Russian Science Foundation under grant 19-11-00001.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Ol’mezov, K.I., Semchenkov, A.S. & Shkredov, I.D. On Popular Sums and Differences for Sets with Small Multiplicative Doubling. Math Notes 108, 557–565 (2020). https://doi.org/10.1134/S000143462009028X
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1134/S000143462009028X