Besides various asymptotic results on the concept of sum-product bases in $\mathbb{N}_0$, we consider by probabilistic arguments the existence of thin sets $A,A'$ of integers such that $AA+A=\mathbb{N}_0$ and $A'A'+A'A'=\mathbb{N}_0$.

In this paper we establish new estimates on sum-product sets and certain exponential sums in finite fields of prime order. Our first result is an extension of the sum-product theorem from [8] when… Expand

In this paper we investigate how small the density of a multiplicative basis of order h can be in {1,2,...,n} and in ℤ+. Furthermore, a related problem of Erdős is also studied: How dense can a set… Expand

Let $$F(x,y,z)=xy+z$$F(x,y,z)=xy+z. We consider some properties of expansion of the polynomial F in different settings, namely in the integers and in prime fields. The main results concern the… Expand

The circle method is introduced, which is a nice application of Fourier analytic techniques to additive problems and its other applications: Vinogradov without GRH, partitions, Waring’s problem.Expand