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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
Acta Mathematica Academiae Scientiarum Hungarica – Springer Journals
Published: May 21, 2016
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