Let $0<r<1$ a real number which is not a fraction of the form $p/2^n$ for any integers $p,n$. Now, for every integer $n\ge 1$ we can find the *closest* fraction of the form $p/2^n$ to $r$, which will be on the left or on the right of $r$. For example, for $r=2/3$, those fractions will be: $1/2$, $3/4$, $5/8$, $11/16$ etc.

We will now look at the sequence $a_n=(3/2)^n\pmod 1$, which itself contains fractions of the form $p/2^n$ from the interval $(0,1)$, and could, for any given $r$, potentially "hit" one of those "closest" fractions to $r$.

Now, my question is this: can it be proven that, *for every* $r$, $0<r<1$, not of the form $p/2^n$, the sequence $a_n$ above contains at least one of those "closest" fractions to $r$? It seems intuitive to me that this should be true.