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Linear problems in combinatorial number theory

Linear problems in combinatorial number theory Acta Matkematica Academiae Scientiarum Hungaricae Tomus 26 (1--2), (1975), 113--121. LINEAR PROBLEMS IN COMBINATORIAL NUMBER THEORY By J. KOMLOS, M. SULYOK and E. SZEMERt~DI (Budapest) Introduction In many problems of combinatorial number theory one has to estimate the density of a sequence of positive integers, where the only restriction on the sequence is that some linear relations do not hold for its subsets. We mention two such problems which have been investigated by many authors. The first one is the Sidon-problem. A sequence a~ (finite or infinite) is called a B~-sequence if every integer can be written in at most one way in the form ai+aj., In general, a sequence is a B~-sequence if the sums a~+... +a~ are all different. Let Fh(n) denote the largest number k for which there exists a Bh sequence a~<a2<... <ak<=n. Sidon applied the estimations on Fz(n) for Fourier-analysis. Erd6s, Tudm, R6nyi, Singer, Chowla and many others investigated the asymptotic behaviour of the functions Fh (n) and proved that F2(n)~[/n and c~nW'<F~(n)<c2n 1lb. (Further base-problems can be found in [1].) The following problem is similar in character, but is much harder: How many numbers can be chosen from the first n integers not http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Acta Mathematica Academiae Scientiarum Hungarica Springer Journals

Linear problems in combinatorial number theory

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Publisher
Springer Journals
Copyright
Copyright
Subject
Mathematics; Mathematics, general
ISSN
0001-5954
eISSN
1588-2632
DOI
10.1007/BF01895954
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Abstract

Acta Matkematica Academiae Scientiarum Hungaricae Tomus 26 (1--2), (1975), 113--121. LINEAR PROBLEMS IN COMBINATORIAL NUMBER THEORY By J. KOMLOS, M. SULYOK and E. SZEMERt~DI (Budapest) Introduction In many problems of combinatorial number theory one has to estimate the density of a sequence of positive integers, where the only restriction on the sequence is that some linear relations do not hold for its subsets. We mention two such problems which have been investigated by many authors. The first one is the Sidon-problem. A sequence a~ (finite or infinite) is called a B~-sequence if every integer can be written in at most one way in the form ai+aj., In general, a sequence is a B~-sequence if the sums a~+... +a~ are all different. Let Fh(n) denote the largest number k for which there exists a Bh sequence a~<a2<... <ak<=n. Sidon applied the estimations on Fz(n) for Fourier-analysis. Erd6s, Tudm, R6nyi, Singer, Chowla and many others investigated the asymptotic behaviour of the functions Fh (n) and proved that F2(n)~[/n and c~nW'<F~(n)<c2n 1lb. (Further base-problems can be found in [1].) The following problem is similar in character, but is much harder: How many numbers can be chosen from the first n integers not

Journal

Acta Mathematica Academiae Scientiarum Hungarica – Springer Journals

Published: May 21, 2016

References

  • Sequences
    Halberstam, H.; Roth, K. F.
  • On certain sets of integers, II
    Roth, K. F.
  • On sets of integers which contain no three terms in arithmetical progression
    Behrend, F. A.
  • On sets of integers containing no four elements in arithmetic progression
    Szemerédi, E.