Counting and Hausdorff measures for integers and $p$-adic integers

In this work, we aim to advance the development of a fractal theory for sets of integers. The core idea is to utilize the fractal structure of $p$-adic integers, where $p$ is a prime number, and compare this with conventional densities and counting measures for integers. Our approach yields some results in combinatorial number theory. The results show how the local fractal structure of a set in $\mathbb{Z}_p$ can provide bounds for the counting measure for its projection onto $\mathbb{Z}$. Additionally, we establish a relationship between the counting dimension of a set of integers and its box-counting dimension in $\mathbb{Z}_p$. Since our results pertain to sets that are projections of closed sets in $\mathbb{Z}_p$, we also provide both necessary and sufficient combinatorial conditions for a set $E\subset \mathbb{Z}$ to be the projection of a closed set in $\mathbb{Z}_p$.