Let $f:\mathbb{N}\to (0,\infty )$ be a weight function. For a subset $A\subset \mathbb{N}$ let

Define

the lower an upper $f$-densities of $A$, respectively. In the case when ${\underline{d}}_f\left(A\right)={\overline{d}}_f\left(A\right)$ we say that $A$ possesses $f$-density $d_f\left(A\right)$. A weight function $f$ is called regular if $d_f(a\mathbb{N}+b)=\frac{1}{a}$ for every natural numbers $a,b\ge 1$. Let $f,g$ be regular weight functions. Let $B\subset \mathbb{N}$ such that

Let $0\le \alpha \le \beta \le \gamma \le \delta \le 1$. The authors prove that there exists $A\subset \mathbb{N}$ such that

Let the functions $f,g:\mathbb{N}\to (0,\infty )$ satisfy the assumptions

Then there exists $B\subset \mathbb{N}$ such that