Consider the following (easy) lemma.

Lemma. There are a subset $Q$ of the positive integers and a constant $N > 0$ such that  $Q$ has positive asymptotic density and for each rational numbers $\alpha,\beta$ it results $\alpha \beta^n \in Q$ for at most $N$ positive integers $n$.

Proof (sketch). Take $Q$ as the set of squarefree positive integers and $N=2$.

My question is: For a fixed rational number $r$, can we replace $\alpha \beta^n$ with $\alpha \beta^n + r$ in the lemma above?

Thank you in advance for any help.

Consider the following (easy) lemma.

Lemma. There is a subset $Q$ of the positive integers and a fixed constant $N > 0$ such that
1)$Q$ has positive asymptotic density and

2)for each rational numbers $\alpha,\beta$ it results $\alpha \beta^n \in Q$ for at most $N$ positive integers $n$.

Proof (sketch). Take $Q$ as the set of squarefree positive integers and $N=2$.

My question is: For a fixed rational number $r$, can we replace $\alpha \beta^n$ with $\alpha \beta^n + r$ in the lemma above?

Thank you in advance for any help.