Skip to main content
Log in

On an Application of Higher Energies to Sidon Sets

Combinatorica Aims and scope Submit manuscript

Cite this article


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 \(|A|^{1/2+c}\).

This is a preview of subscription content, log in via an institution to check access.

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions


  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 

Download references

Author information

Authors and Affiliations


Corresponding author

Correspondence to I. D. Shkredov.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

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

Rights and permissions

Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Reprints and Permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

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

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI:


Mathematics Subject Classification