# On sum-product bases

## TL;DR

This paper explores the concept of sum-product bases in non-negative integers, proving the existence of thin sets that can generate all non-negative integers through sum and product operations using probabilistic methods.

## Contribution

It introduces new probabilistic techniques to demonstrate the existence of thin sum-product bases that cover all non-negative integers.

## Key findings

- Existence of thin sets A, A' such that AA + A = N_0
- Existence of sets A', A' with A'A' + A'A' = N_0
- Use of probabilistic arguments to establish these results

## Abstract

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 $A'A'+A'A'=\mathbb{N}_0$.