Problems in additive number theory

In the first chapter we obtain the Biro-type upper bound for the smallest period of B in the case when A is a finite multiset of integers and B is a multiset such that A and B are t-complementing multisets of integers. In the second chapter we answer an inverse problem for lattice points proving that if K is a compact subset of RxR such that K+ZxZ=RxR then the integer points of the difference set of K is not contained on the coordinate axes, Zx{0}∪{0}xZ. In the third chapter we show that there exist infinite sets A and M of positive integers whose partition function has weakly superpolynomial but not superpolynomial growth. The last chapter deals with the size of a sum of dilates 2·A+k·A. We prove that if k is a power of an odd prime or product of two primes and A a finite set of integers such that |A|>8kk,then |2· A+k·A|≥ ( k+2)|A|-k2-k+2.