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ROTH'S THEOREM FOR FOUR VARIABLES AND ADDITIVE STRUCTURES IN SUMS OF SPARSE SETS
KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).ORCID iD: 0000-0003-4273-593X
2016 (English)In: FORUM OF MATHEMATICS SIGMA, ISSN 2050-5094, Vol. 4, UNSP e5Article in journal (Refereed) Published
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Abstract [en]

We show that if A subset of {1,...,N} does not contain any nontrivial solutions to the equation x + y + z = 3w, then vertical bar A vertical bar <= N/exp(c(log N)(1/7))' 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 of {1, ...,N}, or high-dimensional affine subspaces if A subset of F-q(n), even if A has density of the shape above.

Place, publisher, year, edition, pages
Cambridge University Press, 2016. Vol. 4, UNSP e5
National Category
Mathematics
Identifiers
URN: urn:nbn:se:kth:diva-185077DOI: 10.1017/fms.2016.2ISI: 000372351600001OAI: oai:DiVA.org:kth-185077DiVA: diva2:919744
Note

QC 20160414

Available from: 2016-04-14 Created: 2016-04-11 Last updated: 2016-04-14Bibliographically approved

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