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On an Application of Higher Energies to Sidon Sets

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We show that for any finite set A and an arbitrary \(\varepsilon >0\) there exists \(k=k(\varepsilon )\) such that the higher energy \(\textsf{E} _k(A)\) is at most \(|A|^{k+\varepsilon }\) unless A has a very specific structure. As an application we obtain that any finite subset A of the real numbers or the prime field either contains an additive Sidon-type subset of size \(|A|^{1/2+c}\) or a multiplicative Sidon-type subset of size \(|A|^{1/2+c}\).

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  1. Alon, N., Erdős, P.: An application of graph theory to additive number theory. Eur. J. Comb. 6(3), 201–203 (1985)

    Article  MathSciNet  MATH  Google Scholar 

  2. Brown, W.G.: On graphs that do not contain a Thomsen graph. Can. Math. Bull. 9(3), 281–285 (1966)

    Article  MathSciNet  MATH  Google Scholar 

  3. Cilleruelo, J.: Gaps in dense Sidon sets, Integers: Paper-A11 (2000)

  4. Cilleruelo, J., Ruzsa, I.Z., Vinuesa, C.: Generalized Sidon sets. Adv. Math. 225(5), 2786–2807 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  5. Dubickas, A., Schoen, T., Silva, M., Sarka, P.: Finding large co-Sidon subsets in sets with a given additive energy. Eur. J. Comb. 34(7), 1144–1157 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  6. Erdős, P.: Extremal problems in number theory, combinatorics and geometry, Proc. Inter. Congress in Warsaw (1983)

  7. Erdős, P., Harzheim, E.: Congruent subsets of infinite sets of natural numbers. J. Reine Angew. Math. 367, 207–214 (1986)

    MathSciNet  MATH  Google Scholar 

  8. Erdős, P., Turán, P.: On a problem of Sidon in additive number theory, and on some related problems. J. Lond. Math. Soc. 16(4), 212–215 (1941)

    Article  MathSciNet  MATH  Google Scholar 

  9. Füredi, Z., Simonovits, M.: The History of Degenerate (Bipartite) Extremal Graph Problems, pp. 169–264. Erdős Centennial. Springer, Berlin, Heidelberg (2013)

    MATH  Google Scholar 

  10. Green, B.: The number of squares and \(B_h[g]\) sets. Acta Arith. 100(4), 365–390 (2001)

    Article  MathSciNet  MATH  Google Scholar 


  12. Komlós, J., Sulyok, M., Szemerédi, E.: Linear problems in combinatorial number theory. Acta Mathematica Academiae Scientiarum Hungarica 26(1–2), 113–121 (1975)

    Article  MathSciNet  MATH  Google Scholar 

  13. Linström, B.: \(B_h[g]\)-sequences from \(B_h\)-sequences. Proc. Am. Math. Soc. 128, 657–659 (2000)

    Article  MATH  Google Scholar 

  14. Linström, B.: An inequality for \(B_2\)-sequences. J. Comb. Theory 6(2), 211–212 (1969)

    Article  Google Scholar 

  15. Murphy, B.: Upper and lower bounds for rich lines in grids. (2017)

  16. Nathanson, M.B.: Every function is the representation function of an additive basis for the integers. Portugaliae Mathematica. Nova Série 62(1), 55–72 (2005)

    MathSciNet  MATH  Google Scholar 

  17. O’Bryant, K.: A complete annotated bibliography of work related to Sidon sequences. Electron. J. Combin. Dynamic Surveys, Paper No. DS11, p. 39. (2004)

  18. Peng, X., Tesoro, R., Timmons, C.: Bounds for generalized Sidon sets. Discret. Math. 338(3), 183–190 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  19. Roche–Newton, O., Warren, A.: Additive and multiplicative Sidon sets.

  20. Ruzsa, I.Z.: An infinite Sidon sequence. J. Number Theory 68(1), 63–71 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  21. Schoen, T., Shkredov, I.D.: Higher moments of convolutions. J. Number Theory 133(5), 1693–1737 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  22. Semchankau, A.S.: Maximal subsets free of arithmetic progressions in arbitrary sets. Math. Notes 102(3), 396–402 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  23. Shkredov, I.D.: Energies and structure of additive sets. Electron. J. Comb. 21, 3 (2014)

    MathSciNet  MATH  Google Scholar 

  24. Shkredov, I.D.: Some remarks on the Balog-Wooley decomposition theorem and quantities \(D^+\), \(D^\times \). Proc. Steklov Inst. Math. 298(1), 74–90 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  25. Shkredov, I.D.: Modular hyperbolas and bilinear forms of Kloosterman sums. J. Number Theory 220, 182–211 (2021)

    Article  MathSciNet  MATH  Google Scholar 

  26. Sidon, S.: Ein Satz über trigonometrische Polynome und seine Anwendungen in der Theorie der Fourier-Reihen. Math. Annalen 106, 536–539 (1932)

    Article  MathSciNet  MATH  Google Scholar 

  27. Tao, T., Vu, V.: Additive Combinatorics. Cambridge University Press, Cambridge (2006)

    Book  MATH  Google Scholar 

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Correspondence to I. D. Shkredov.

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This work is supported by the Russian Science Foundation under Grant 19–11–00001, ject/19-11-00001.

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Shkredov, I.D. On an Application of Higher Energies to Sidon Sets. Combinatorica 43, 329–345 (2023).

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