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ROTH’S THEOREM FOR FOUR VARIABLES AND ADDITIVE STRUCTURES IN SUMS OF SPARSE SETS

  • TOMASZ SCHOEN (a1) and OLOF SISASK (a2)

Abstract

We show that if $A\subset \{1,\ldots ,N\}$ does not contain any nontrivial solutions to the equation $x+y+z=3w$ , then

$$\begin{eqnarray}|A|\leqslant \frac{N}{\exp (c(\log N)^{1/7})},\end{eqnarray}$$
where $c>0$ is some absolute constant. In view of Behrend’s construction, this bound is of the right shape: the exponent $1/7$ cannot be replaced by any constant larger than $1/2$ . We also establish a related result, which says that sumsets $A+A+A$ contain long arithmetic progressions if $A\subset \{1,\ldots ,N\}$ , or high-dimensional affine subspaces if $A\subset \mathbb{F}_{q}^{n}$ , even if $A$ has density of the shape above.

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Copyright

This is an Open Access article, distributed under the terms of the Creative Commons Attribution-NonCommercial-ShareAlike licence (http://creativecommons.org/licenses/by-nc-sa/4.0/), which permits non-commercial re-use, distribution, and reproduction in any medium, provided the same Creative Commons licence is included and the original work is properly cited. The written permission of Cambridge University Press must be obtained for commercial re-use.

References

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[1] Bateman, M. and Katz, N. H., ‘New bounds on cap sets’, J. Amer. Math. Soc. 25(2) (2012), 585613. arXiv:1101.5851.
[2] Behrend, F. A., ‘On sets of integers which contain no three terms in arithmetical progression’, Proc. Natl. Acad. Sci. USA 32 (1946), 331332.
[3] Bloom, T. F., ‘Translation invariant equations and the method of Sanders’, Bull. Lond. Math. Soc. 44(5) (2012), 10501067. arXiv:1107.1110.
[4] Bloom, T. F., A quantitative improvement for Roth’s theorem on arithmetic progressions,arXiv:1405.5800.
[5] Bourgain, J., ‘On triples in arithmetic progression’, Geom. Funct. Anal. 9(5) (1999), 968984.
[6] Bourgain, J., ‘Roth’s theorem on progressions revisited’, J. Anal. Math. 104 (2008), 155192.
[7] Croot, E., Łaba, I. and Sisask, O., ‘Arithmetic progressions in sumsets and L p -almost-periodicity’, Combin. Probab. Comput. 22(3) (2013), 351365. arXiv:1103.6000.
[8] Croot, E., Ruzsa, I. Z. and Schoen, T., ‘Arithmetic progressions in sparse sumsets’, inCombinatorial Number Theory (de Gruyter, Berlin, 2007), 157164.
[9] Croot, E. and Sisask, O., ‘A new proof of Roth’s theorem on arithmetic progressions’, Proc. Amer. Math. Soc. 137 (2009), 805809. arXiv:0801.2577.
[10] Croot, E. and Sisask, O., ‘A probabilistic technique for finding almost-periods of convolutions’, Geom. Funct. Anal. 20(6) (2010), 13671396. arXiv:1003.2978.
[11] Croot, E. and Sisask, O., Notes on proving Roth’s theorem using Bogolyubov’s method, http://people.math.gatech.edu/∼ecroot/bogolyubov-roth2.pdf.
[12] Elkin, M., ‘An improved construction of progression-free sets’, Israel J. Math. 184 (2011), 93128. arXiv:0801.4310.
[13] Freiman, G. A., Halberstam, H. and Ruzsa, I. Z., ‘Integer sum sets containing long arithmetic progressions’, J. Lond. Math. Soc. 46(2) (1992), 193201.
[14] Green, B., ‘Arithmetic progressions in sumsets’, Geom. Funct. Anal. 12(3) (2002), 584597.
[15] Green, B., ‘Finite field models in additive combinatorics’, inSurveys in Combinatorics 2005, London Mathematical Society Lecture Note Series, 327 (Cambridge University Press, Cambridge, 2005), 127. arXiv:math/0409420.
[16] Green, B. and Ruzsa, I. Z., ‘Freiman’s theorem in an arbitrary abelian group’, J. Lond. Math. Soc. (2) 75(1) (2007), 163175. arXiv:math/0505198.
[17] Green, B. and Wolf, J., ‘A note on Elkin’s improvement of Behrend’s construction’, inAdditive Number Theory (Springer, New York, 2010), 141144. arXiv:0810.0732.
[18] Heath-Brown, D. R., ‘Integer sets containing no arithmetic progressions’, J. Lond. Math. Soc. (2) 35(3) (1987), 385394.
[19] Henriot, K., ‘On arithmetic progressions in A + B + C ’, Int. Math. Res. Notices 2014(18) (2014), 51345164. arXiv:1211.4917.
[20] Roth, K. F., ‘On certain sets of integers’, J. Lond. Math. Soc. 28 (1953), 104109.
[21] Sanders, T., ‘Additive structures in sumsets’, Math. Proc. Cambridge Philos. Soc. 144(2) (2008), 289316. arXiv:math/0605520.
[22] Sanders, T., ‘Green’s sumset problem at density one half’, Acta Arith. 146(1) (2011), 91101. arXiv:1003.5649.
[23] Sanders, T., ‘On Roth’s theorem on progressions’, Ann. of Math. (2) 174(1) (2011), 619636. arXiv:1011.0104.
[24] Sanders, T., ‘On certain other sets of integers’, J. Anal. Math. 116 (2012), 5382. arXiv:1007.5444.
[25] Sanders, T., ‘On the Bogolyubov-Ruzsa lemma’, Anal. PDE 5(3) (2012), 627655. arXiv:1011.0107.
[26] Schoen, T. and Shkredov, I., ‘Roth’s theorem in many variables’, Israel J. Math. 199(1) (2014), 287308. arXiv:1106.1601.
[27] Szemerédi, E., ‘Integer sets containing no arithmetic progressions’, Acta Math. Hungar. 56(1–2) (1990), 155158.
[28] Tao, T. and Vu, V. H., Additive Combinatorics (Cambridge University Press, Cambridge, 2006).
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ROTH’S THEOREM FOR FOUR VARIABLES AND ADDITIVE STRUCTURES IN SUMS OF SPARSE SETS

  • TOMASZ SCHOEN (a1) and OLOF SISASK (a2)

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