Rectification Principles in Additive Number Theory

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_ k+2 for sufficiently large k? Conway and Guy showed that the answer is affirmative (unpublished) .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

Three Topics in Additive Prime Number Theory

We discuss, in varying degrees of detail, three contemporary themes in prime number theory. Topic 1: the work of Goldston, Pintz and Yıldırım on short gaps between primes. Topic 2: the work of Mauduit and Rivat, establishing that 50% of the primes have odd digit sum in base 2. Topic 3: work of Tao and the author on linear equations in primes. Introduction. These notes are to accompany two lectu...