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January  2021, 20(1): 369-388. doi: 10.3934/cpaa.2020271

On optimal autocorrelation inequalities on the real line

José Madrid 1,, and  João P. G. Ramos 2,

1. 

Department of Mathematics, University of California, Los Angeles, Portola Plaza 520, Los Angeles, California, 90095, USA
2. 

Instituto Nacional de Matemática Pura e Aplicada, Estrada Dona Castorina 110, Rio de Janeiro, RJ, 22460-320, Brazil

*Corresponding author

Received  August 2020 Revised  September 2020 Published  November 2020

We study autocorrelation inequalities, in the spirit of Barnard and Steinerberger's work [1]. In particular, we obtain improvements on the sharp constants in some of the inequalities previously considered by these authors, and also prove existence of extremizers to these inequalities in certain specific settings. Our methods consist of relating the inequalities in question to other classical sharp inequalities in Fourier analysis, such as the sharp Hausdorff–Young inequality, and employing functional analysis as well as measure theory tools in connection to a suitable dual version of the problem to identify and impose conditions on extremizers.

Keywords: autocorrelation, autoconvolution, sharp inequality, extremizers.
Mathematics Subject Classification: 42A05, 42A85, 28A12, 42A82.
Citation: José Madrid, João P. G. Ramos. On optimal autocorrelation inequalities on the real line. Communications on Pure & Applied Analysis, 2021, 20 (1) : 369-388. doi: 10.3934/cpaa.2020271
References:
[1]

R.C. Barnard and S. Steinerberger, Three convolution inequalities on the real line with connection to additive combinatorics, J. Number Theory, 207 (2020), 42-55.  doi: 10.1016/j.jnt.2019.07.001.  Google Scholar

[2]

W. Beckner, Inequalities in Fourier analysis, Ann. Math., 102 (1975), 159-182.  doi: 10.2307/1970980.  Google Scholar

[3]

J. Bourgain, L. Clozel and J.P. Kahane, Principe d'Heisenberg et fonctions positives, Annales de l'institut Fourier, 60 (2010), 1215-1232.   Google Scholar

[4]

J. Cilleruelo, I. Ruzsa and C. Trujillo, Upper and lower bounds for finite $Bh[g]$ sequences, J. Number Theory, 97 (2002), 26-34.  doi: 10.1006/jnth.2001.2767.  Google Scholar

[5]

J. Cilleruelo, I. Ruzsa and C. Vinuesa, Generalized Sidon sets, Adv. Math., 225 (2010), 2786 –2807. doi: 10.1016/j.aim.2010.05.010.  Google Scholar

[6]

A. Cloninger and S. Steinerberger, On suprema of autoconvolutions with an application to Sidon sets, P. Am. Math. Soc., 145 (2017), 3191-3200.  doi: 10.1090/proc/13690.  Google Scholar

[7]

H. Cohn and F. Gonçalves, An optimal uncertainty principle in twelve dimensions via modular forms, Inventionnes Mathematicae, 217 (2019), 799-831.  doi: 10.1007/s00222-019-00875-4.  Google Scholar

[8]

S. Fish, D. King and S. J. Miller, Extensions of Autocorrelation Inequalities with Applications to Additive Combinatorics, arXiv: 2001.02326. Google Scholar

[9] G. B. Folland, A Course in Abstract Harmonic Analysis, CRC Press, 2016.   Google Scholar
[10]

F. Gonçalves, D. Oliveira e Silva and J. P. G. Ramos, New sign uncertainty principles, Preprint, 2020. Google Scholar

[11]

F. Gonçalves, D. Oliveira e Silva and J. P. G. Ramos, On regularity and mass concentration phenomena for the sign uncertainty principle, Preprint, 2020. Google Scholar

[12]

F. Gonçalves, D. Oliveira e Silva and S. Steinerberger, Hermite polynomials, linear flows on the torus, and an uncertainty principle for roots, J. Math. Anal. Appl., 451 (2017), 678-711.  doi: 10.1016/j.jmaa.2017.02.030.  Google Scholar

[13]

B. Green, The number of squares and $Bh[g]$ sets, Acta Arith., 100 (2001), 365-390.  doi: 10.4064/aa100-4-6.  Google Scholar

[14]

G. Martin and K. O'Bryant, Constructions of generalized Sidon sets, J. Comb. Theory A, 113 (2006), 591-607.  doi: 10.1016/j.jcta.2005.04.011.  Google Scholar

[15]

G. Martin and K. O'Bryant, The symmetric subset problem in continuous Ramsey theory, Exp. Math., 16 (2007), 145-166.   Google Scholar

[16]

M. Matolcsi and C. Vinuesa, Improved bounds on the supremum of autoconvolutions, J. Math. Anal. Appl., 372 (2010), 439-447.  doi: 10.1016/j.jmaa.2010.07.030.  Google Scholar

[17]

G. Yu, An upper bound for $B2[g]$ sets, J. Number Theory, 122 (2007), 211-220.  doi: 10.1016/j.jnt.2006.04.008.  Google Scholar

show all references

References:
[1]

R.C. Barnard and S. Steinerberger, Three convolution inequalities on the real line with connection to additive combinatorics, J. Number Theory, 207 (2020), 42-55.  doi: 10.1016/j.jnt.2019.07.001.  Google Scholar

[2]

W. Beckner, Inequalities in Fourier analysis, Ann. Math., 102 (1975), 159-182.  doi: 10.2307/1970980.  Google Scholar

[3]

J. Bourgain, L. Clozel and J.P. Kahane, Principe d'Heisenberg et fonctions positives, Annales de l'institut Fourier, 60 (2010), 1215-1232.   Google Scholar

[4]

J. Cilleruelo, I. Ruzsa and C. Trujillo, Upper and lower bounds for finite $Bh[g]$ sequences, J. Number Theory, 97 (2002), 26-34.  doi: 10.1006/jnth.2001.2767.  Google Scholar

[5]

J. Cilleruelo, I. Ruzsa and C. Vinuesa, Generalized Sidon sets, Adv. Math., 225 (2010), 2786 –2807. doi: 10.1016/j.aim.2010.05.010.  Google Scholar

[6]

A. Cloninger and S. Steinerberger, On suprema of autoconvolutions with an application to Sidon sets, P. Am. Math. Soc., 145 (2017), 3191-3200.  doi: 10.1090/proc/13690.  Google Scholar

[7]

H. Cohn and F. Gonçalves, An optimal uncertainty principle in twelve dimensions via modular forms, Inventionnes Mathematicae, 217 (2019), 799-831.  doi: 10.1007/s00222-019-00875-4.  Google Scholar

[8]

S. Fish, D. King and S. J. Miller, Extensions of Autocorrelation Inequalities with Applications to Additive Combinatorics, arXiv: 2001.02326. Google Scholar

[9] G. B. Folland, A Course in Abstract Harmonic Analysis, CRC Press, 2016.   Google Scholar
[10]

F. Gonçalves, D. Oliveira e Silva and J. P. G. Ramos, New sign uncertainty principles, Preprint, 2020. Google Scholar

[11]

F. Gonçalves, D. Oliveira e Silva and J. P. G. Ramos, On regularity and mass concentration phenomena for the sign uncertainty principle, Preprint, 2020. Google Scholar

[12]

F. Gonçalves, D. Oliveira e Silva and S. Steinerberger, Hermite polynomials, linear flows on the torus, and an uncertainty principle for roots, J. Math. Anal. Appl., 451 (2017), 678-711.  doi: 10.1016/j.jmaa.2017.02.030.  Google Scholar

[13]

B. Green, The number of squares and $Bh[g]$ sets, Acta Arith., 100 (2001), 365-390.  doi: 10.4064/aa100-4-6.  Google Scholar

[14]

G. Martin and K. O'Bryant, Constructions of generalized Sidon sets, J. Comb. Theory A, 113 (2006), 591-607.  doi: 10.1016/j.jcta.2005.04.011.  Google Scholar

[15]

G. Martin and K. O'Bryant, The symmetric subset problem in continuous Ramsey theory, Exp. Math., 16 (2007), 145-166.   Google Scholar

[16]

M. Matolcsi and C. Vinuesa, Improved bounds on the supremum of autoconvolutions, J. Math. Anal. Appl., 372 (2010), 439-447.  doi: 10.1016/j.jmaa.2010.07.030.  Google Scholar

[17]

G. Yu, An upper bound for $B2[g]$ sets, J. Number Theory, 122 (2007), 211-220.  doi: 10.1016/j.jnt.2006.04.008.  Google Scholar

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