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ADDITIVE DIMENSION AND A THEOREM OF SANDERS

Part of: Sequences and sets

Published online by Cambridge University Press:  22 October 2015

TOMASZ SCHOEN  and
ILYA D. SHKREDOV
TOMASZ SCHOEN
Affiliation:
Faculty of Mathematics and Computer Science, Adam Mickiewicz University, Umultowska 87, 61-614 Poznań, Poland email [email protected]
ILYA D. SHKREDOV*
Affiliation:
Division of Algebra and Number Theory, Steklov Mathematical Institute, ul. Gubkina, 8, Moscow, 119991, Russia Delone Laboratory of Discrete and Computational Geometry, Yaroslavl State University, Sovetskaya str. 14, Yaroslavl, 150000, Russia IITP RAS, Bolshoy Karetny per. 19, Moscow, 127994, Russia email [email protected]
*
✉[email protected]
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Abstract

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We prove some new bounds for the size of the maximal dissociated subset of structured (having small sumset, large energy and so on) subsets of an abelian group.

Keywords

additive dimensionsumsetadditive energydissociated sets

MSC classification

Primary: 11B13: Additive bases, including sumsets
Secondary: 11B75: Other combinatorial number theory
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 100 , Issue 1 , February 2016 , pp. 124 - 144
Copyright
© 2015 Australian Mathematical Publishing Association Inc. 

References

Bateman, M. and Katz, N., ‘Structure in additively nonsmoothing sets’, Preprint, 2011, arXiv:1104.2862v1.Google Scholar
Bateman, M. and Katz, N., ‘New bounds on cap sets’, J. Amer. Math. Soc. 25(2) (2012), 585–613.CrossRefGoogle Scholar
Bourgain, J., ‘On arithmetic progressions in sums of sets of integers’, in: A Tribute of Paul Erdös (Cambridge University Press, Cambridge, 1990), 105–109.CrossRefGoogle Scholar
Bourgain, J., ‘On triples in arithmetic progression’, Geom. Funct. Anal. 9 (1999), 968–984.CrossRefGoogle Scholar
Bourgain, J., ‘Roth’s theorem on progressions revisited’, J. Anal. Math. 104 (2008), 155–206.CrossRefGoogle Scholar
Bukh, B., ‘Sums of dilates’, Combin. Probab. Comput. 17(5) (2008), 627–639.CrossRefGoogle Scholar
Candela, P. and Helfgott, H. A., ‘On the dimension of additive sets’, Acta Arith. 167(1) (2015), 91–100.CrossRefGoogle Scholar
Chang, M.-C., ‘A polynomial bound in Freiman’s theorem’, Duke Math. J. 113(3) (2002), 399–419.CrossRefGoogle Scholar
Green, B. and Tao, T., ‘An equivalence between inverse sumset theorems and inverse conjectures for the U 3 -norms’, Math. Proc. Cambridge Philos. Soc. 149(1) (2010), 1–19.CrossRefGoogle Scholar
Lev, V. F. and Yuster, R., ‘On the size of dissociated bases’, Electron. J. Combin. 18(1) (2011), #P117.CrossRefGoogle Scholar
Łuczak, T. and Schoen, T., ‘On a problem of Konyagin’, Acta Arith. 134(2) (2008), 101–109.CrossRefGoogle Scholar
Rudin, W., Fourier Analysis on Groups (John Wiley & Sons Inc., New York, 1990), (reprint of the 1962 original).CrossRefGoogle Scholar
Ruzsa, I. Z., ‘Solving linear equations in sets of integers. I’, Acta Arith. 65 (1993), 259–282.CrossRefGoogle Scholar
Ruzsa, I. Z., ‘Cardinality questions about sumsets’, in: Additive Combinatorics, CRM Proceedings & Lecture Notes, 43 (American Mathematical Society, Providence, RI, 2007), 195–205.CrossRefGoogle Scholar
Ruzsa, I. Z., ‘Sumsets and structure’, in: Combinatorial Number Theory and Additive Group Theory (Birkhäuser, Basel, 2009), 87–210.Google Scholar
Sanders, T., ‘On a theorem of Shkredov’, Online J. Anal. Combin. (5) (2010), Art. 5, 4 pp.Google Scholar
Sanders, T., ‘On Roth’s theorem on progressions’, Ann. of Math. (2) 174 (2011), 619–636.CrossRefGoogle Scholar
Sanders, T., ‘On the Bogolubov–Ruzsa lemma’, Anal. PDE 5 (2012), 627–655.CrossRefGoogle Scholar
Sanders, T., ‘On certain other sets of integers’, J. Anal. Math. 116 (2012), 53–82.CrossRefGoogle Scholar
Sanders, T., ‘Structure in sets with logarithmic doubling’, Canad. Math. Bull. 56(2) (2013), 412–423.CrossRefGoogle Scholar
Sanders, T., ‘The structure theory of set addition revisited’, Bull. Amer. Math. Soc. (N.S.) 50(1) (2013), 93–127.CrossRefGoogle Scholar
Schoen, T., ‘Near optimal bounds in Freiman’s theorem’, Duke Math. J. 158 (2011), 1–12.CrossRefGoogle Scholar
Schoen, T. and Shkredov, I. D., ‘Higher moments of convolutions’, J. Number Theory 133 (2013), 1693–1737.CrossRefGoogle Scholar
Shkredov, I. D., ‘On sets with small doubling’, Mat. Zametki 84(6) (2008), 927–947.Google Scholar
Shkredov, I. D., ‘Some new results on higher energies’, Trans. Moscow Math. Soc. 74(1) (2013), 35–73.Google Scholar
Shkredov, I. D. and V’ugin, I. V., ‘On additive shifts of multiplicative subgroups’, Mat. Sb. 203(6) (2012), 81–100.Google Scholar
Shkredov, I. D. and Yekhanin, S., ‘Sets with large additive energy and symmetric sets’, J. Combin. Theory Ser. A 118 (2011), 1086–1093.CrossRefGoogle Scholar
Tao, T. and Vu, V., Additive Combinatorics (Cambridge University Press, Cambridge, 2006).CrossRefGoogle Scholar