Hostname: page-component-586b7cd67f-gb8f7 Total loading time: 0 Render date: 2024-11-30T15:41:37.932Z Has data issue: false hasContentIssue false
  • English
  • Français

On Convolutions of Convex Sets and Related Problems

Published online by Cambridge University Press:  20 November 2018

Tomasz Schoen
Tomasz Schoen*
Affiliation:
Faculty ofMathematics and Computer Science, AdamMickiewiczUniversity, Umultowska 87, 61-614 Poznań, Poland e-mail: [email protected]
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

We prove some results concerning convolutions, additive energies, and sumsets of convex sets and their generalizations. In particular, we show that if a set $A\,=\,{{\{{{a}_{1}},\,.\,.\,.\,,\,{{a}_{n}}\}}_{<}}\,\subseteq \,\mathbb{R}$ has the property that for every fixed $1\,\le \,d\,<\,n$, all differences ${{a}_{i}}\,-\,{{a}_{i-d}},\,d\,<\,i\,<n$, are distinct, then $\left| A\,+\,A \right|\,\gg \,{{\left| A \right|}^{3/2+c}}$ for a constant $c\,>\,0$.

Keywords

11B99convex setsadditive energysumsets
Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 57 , Issue 4 , 01 December 2014 , pp. 877 - 883
Copyright
Copyright © Canadian Mathematical Society 2014

References

[1] Bochkarev, S.W., Multiplicative inequalities for L1-norm, applications to analysis and number theory. (Russian) Tr. Mat. Inst. Steklova 255 (2006), Funkts. Prostran., Teor. Priblizh., Nelinein. Anal., 55–70; translation in Proc. Steklov Inst. Math. 2006, no. 4(255), 49–64 .Google Scholar
[2] Elekes, G., Nathanson, M., and Ruzsa, I. Z., Convexity and sumsets. J. Number Theory 83 (2000), no. 2, 194–201. http://dx.doi.org/10.1006/jnth.1999.2386 Google Scholar
[3] Garaev, M. Z., On lower bounds for L1-norm of exponential sums. (Russian) Mat. Zametki 68 (2000), no. 6, 842–850; translation in Math. Notes 68 (2000), no. 5–6, 713–720. http://dx.doi.org/10.4213/mzm1006 Google Scholar
[4] Garaev, M. Z., On a additive representation associated with L1-norm of exponential sum. Rocky Mountain J. Math. 37 (2007), no. 5, 1551–1556. http://dx.doi.org/10.1216/rmjm/1194275934 Google Scholar
[5] Garaev, M. Z., On the number of solutions of Diophantine equation with symmetric entries. J. Number Theory 125 (2007), no. 1, 201–209. http://dx.doi.org/10.1016/j.jnt.2006.09.018 Google Scholar
[6] Garaev, M. Z. and Kueh, K-L., On cardinality of sumsets. J. Aust. Math. Soc. 78 (2005), no. 2, 221–224. http://dx.doi.org/10.1017/S1446788700008041 Google Scholar
[7] Hegyv´ari, N., On consecutive sums in sequences. Acta Math. Hungar. 48 (1986), no. 1–2, 193–200. http://dx.doi.org/10.1007/BF01949064 Google Scholar
[8] Konyagin, V. S., An estimate of L1-norm of an exponential sum. In: The theory of approximations of functions and operators. abstracts of papers of the international conference dedicated to Stechkin’s 80th Anniversay [in Russian]. Ekaterinburg, 2000, pp. 88–89.Google Scholar
[9] Schoen, T. and Shkredov, I. D., Additive properties of multiplicative subgroups of Fp. Q. J. Math. 63 (2012), no. 3, 713–722. http://dx.doi.org/10.1093/qmath/har002 Google Scholar
[10] Schoen, T. and Shkredov, I. D., Higher moments of convolutions. J. Number Theory 133 (2013), no. 5, 1693–1737. http://dx.doi.org/10.1016/j.jnt.2012.10.010 Google Scholar
[11] Schoen, T. and Shkredov, I. D., On sumsets of convex sets. Combin. Probab. Comput. 20 (2011), no. 5, 793–798. http://dx.doi.org/10.1017/S0963548311000277 Google Scholar
[12] Shkredov, I. D., Some new results on higher energies. http://arxiv:1212.6414 Google Scholar
[13] Solymosi, J., Sum versus product. (Spanish) Gac. R. Soc. Mat. Esp. 12 (2009), no. 4, 707–719.Google Scholar
[14] Szemerédi, E. andTrotter, W. T., Extremal problems in discrete geometry. Combinatorica 3 (1983), no. 3–4, 381–392. http://dx.doi.org/10.1007/BF02579194 Google Scholar