  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 $\mathbb{Z}_p$.
