Let $\epsilon > 0$ be some small number and $P > 1$. Let $I$ be an interval. Let $X(P)$ be a set of $\gamma \in [0,1]$ with the property that $$ |P t - \gamma | < \epsilon, $$ for some $t \in I$.
I would like to obtain an upper bound on measure of $X(P)$. The following is my attempt...
$$
\mbox{meas } X(P) \leq \int_{I} \mbox{meas } ( \{ \gamma \in [0,1]: |P t - \gamma | < \epsilon \} ) dt \leq 2 \epsilon \mbox{ meas }I.
$$
I feel like I am doing something wrong, because the bound doesn't depend on $P$. I would appreciate if someone could clarify what is wrong with the argument... Thank you!