Author:
Tomasz Schoen Faculty of Mathematics and Computer Science, Adam Mickiewicz University, Umultowsa 87, 61-614 Poznań, Poland

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Abstract

We prove two results concerning solvability of a linear equation in sets of integers. In particular, it is shown that for every k∊ℕ, there is a noninvariant linear equation in k variables such that if A⫅{1,…,N} has no solution to the equation then , for some absolute constant c>0, provided that N is large enough.

  • [1] Chang, M.-C. 2002 Polynomial bounds in Freiman's theorem Duke Math. J. 113 399419 .

  • [2] Roth, K. F. 1953 On certain sets of integers J. London Math. Soc. 28 104109 .

  • [3] Ruzsa, I. Z. 1993 Solving linear equations in sets of integers. I Acta Arith. 65 259282.

  • [4] Ruzsa, I. Z. 1995 Solving linear equations in sets of integers. II Acta Arith. 72 385397.

  • [5] Schoen, T. 2005 Linear equations in ℤp Bull. London Math. Soc. 37 495501 .

  • [6] Szemerédi, E. 1975 On sets of integers containing no k elements in arithmetic progression Acta Arth. 27 199245.

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Acta Mathematica Hungarica
Language English
Size B5
Year of
Foundation
1950
Volumes
per Year
3
Issues
per Year
6
Founder Magyar Tudományos Akadémia  
Founder's
Address
H-1051 Budapest, Hungary, Széchenyi István tér 9.
Publisher Akadémiai Kiadó
Springer Nature Switzerland AG
Publisher's
Address
H-1117 Budapest, Hungary 1516 Budapest, PO Box 245.
CH-6330 Cham, Switzerland Gewerbestrasse 11.
Responsible
Publisher
Chief Executive Officer, Akadémiai Kiadó
ISSN 0236-5294 (Print)
ISSN 1588-2632 (Online)

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