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Given a subset $\mathcal S\subset \mathbb N\setminus\{0\}$ of (strictly) positive integers, we can consider subsets $A$ of $\mathbb N$ (or $\mathbb Z$) with no differences in $\mathcal S$. Examples: ...

Let F(n) be A000045 i.e. Fibonacci numbers. Here $$ F(n) = F(n-1) + F(n-2), \\ F(0) = 0, F(1) = 1 $$ Let $$ \ell(n) = \left\lfloor\log_2 n\right\rfloor $$ Let $$ T(n, k) = \left\lfloor\frac{n}{2^k}\...