A quantitative improvement for Roth's theorem on arithmetic progressions

Thomas F Bloom

doi:10.1112/jlms/jdw010

A quantitative improvement for Roth's theorem on arithmetic progressions

Thomas F Bloom

Research output: Contribution to journal › Article (Academic Journal) › peer-review

62 Citations (Scopus)

508 Downloads (Pure)

Abstract

We improve the quantitative estimate for Roth's theorem on three-term arithmetic progressions, showing that if A⊂{1,…,N} contains no non-trivial three-term arithmetic progressions, then ∣A∣≪N(loglogN)4/logN. By the same method, we also improve the bounds in the analogous problem over Fq[t] and for the problem of finding long arithmetic progressions in a sumset.

Original language	English
Pages (from-to)	643-663
Number of pages	21
Journal	Journal of the London Mathematical Society
Volume	93
Issue number	3
Early online date	25 Apr 2016
DOIs	https://doi.org/10.1112/jlms/jdw010
Publication status	Published - 1 Jun 2016

Keywords

11B25 (primary)
11B30
11T55 (secondary)

Access to Document

10.1112/jlms/jdw010

Full-text PDF (accepted author manuscript)
This is the author accepted manuscript (AAM). The final published version (version of record) is available online via Oxford University Press at http://dx.doi.rg/10.1112/jlms/jdw010.
Accepted author manuscript, 350 KB

Handle.net

Persistent link

Cite this

@article{352cee8b45e54525b30f19cd9bc6d14c,

title = "A quantitative improvement for Roth's theorem on arithmetic progressions",

abstract = "We improve the quantitative estimate for Roth's theorem on three-term arithmetic progressions, showing that if A⊂\{1,…,N\} contains no non-trivial three-term arithmetic progressions, then ∣A∣≪N(loglogN)4/logN. By the same method, we also improve the bounds in the analogous problem over Fq[t] and for the problem of finding long arithmetic progressions in a sumset.",

keywords = "11B25 (primary), 11B30, 11T55 (secondary)",

author = "Bloom, \{Thomas F\}",

year = "2016",

month = jun,

day = "1",

doi = "10.1112/jlms/jdw010",

language = "English",

volume = "93",

pages = "643--663",

journal = "Journal of the London Mathematical Society",

issn = "0024-6107",

publisher = "John Wiley \& Sons, Ltd.",

number = "3",

}

TY - JOUR

T1 - A quantitative improvement for Roth's theorem on arithmetic progressions

AU - Bloom, Thomas F

PY - 2016/6/1

Y1 - 2016/6/1

N2 - We improve the quantitative estimate for Roth's theorem on three-term arithmetic progressions, showing that if A⊂{1,…,N} contains no non-trivial three-term arithmetic progressions, then ∣A∣≪N(loglogN)4/logN. By the same method, we also improve the bounds in the analogous problem over Fq[t] and for the problem of finding long arithmetic progressions in a sumset.

AB - We improve the quantitative estimate for Roth's theorem on three-term arithmetic progressions, showing that if A⊂{1,…,N} contains no non-trivial three-term arithmetic progressions, then ∣A∣≪N(loglogN)4/logN. By the same method, we also improve the bounds in the analogous problem over Fq[t] and for the problem of finding long arithmetic progressions in a sumset.

KW - 11B25 (primary)

KW - 11B30

KW - 11T55 (secondary)

U2 - 10.1112/jlms/jdw010

DO - 10.1112/jlms/jdw010

M3 - Article (Academic Journal)

SN - 0024-6107

VL - 93

SP - 643

EP - 663

JO - Journal of the London Mathematical Society

JF - Journal of the London Mathematical Society

IS - 3

ER -

A quantitative improvement for Roth's theorem on arithmetic progressions

Abstract

Keywords

Access to Document

Handle.net

Fingerprint

Cite this