A theorem in additive number theory

Some Applications of Ramsey's Theorem to Additive Number Theory

About 50 years ago, Sidon called a sequence of integers A = {a l < a 2 < • } a B, sequence if the number of representations of n as the sum of r or fewer a's is at most k and for some n is exactly k. In particular he was interested in B21) , or, for short, B 2 sequences. For a B 2 sequence the sums a,+ai are all distinct. In 1933 Sidon asked me to find a B 2 sequence for which an increases as s...

Some Problems in Additive Number Theory

(3) f(x) = (log x/log 2) + 0(1)? 1\Mloser and I asked : Is it true that f(2 11) >_ k+2 for sufficiently large k? Conway and Guy showed that the answer is affirmative (unpublished) . P. Erdös, Problems and results in additive number theory, Colloque, Théorie des Nombres, Bruxelles 1955, p . 137 . 2. Let 1 < a 1< . . . < ak <_ x be a sequence of integers so that all the sums ai,+ . . .+ais, i 1 <...

Fourier Analytic Methods in Additive Number Theory

In recent years, analytic methods have become prominent in additive number theory. In particular, finite Fourier analysis is well-suited to solve some problems that are too difficult for purely combinatorial techniques. Among these is Szemerédi’s Theorem, a statement regarding the density of integral sets and the existence of arithmetic progressions in those sets. In this thesis, we give a gene...

An asymptotic formula in additive number theory

1 . Introduction . In his paper [1], Erdös introduced the sequences of positive integers b 1 < b, < . . ., with (b ;, bj ) = 1, for i ~A j, and 'bi 1 < oo . With any such arbitrary sequence of integers, he associated the sequence {di} of all positive integers not divisible by any bj , and he showed that if b1 > 2, there exists a 0 < 1 (independent of the sequence {b i }) such that d i 1 di < d°...

Some Recent Developments in Additive Number Theory