Let $\alpha$ be any positive irrational and $\beta$ be any positive real. We have the following results.

**H. Weyl** (1909): The fractional part of the sequence $\alpha n$ is equidistributed modulo 1.

**I. Vinogradov** (1935): The fractional part of the sequence $\alpha p_n$ is equidistributed modulo 1 where $p_n$ is the $n$-th prime.

**E. Hlawka** (1975): The fractional part of the sequence $\beta \gamma_n$ is equidistributed modulo 1 where $\gamma_n$ is the imaginary part of the $n$-th zero the Riemann zeta function.

The common thing in each of the above three celebrated results is that the sequences are of the form $as_n$ where $a$ is a positive real and $s_n$ has the property that the sequence

$$ \frac{s_1}{s_n}, \frac{s_2}{s_n}, \ldots , \frac{s_{n-1}}{s_n} $$

approaches equidistribution modulo 1 as $n \to \infty$.

**Question**: I would like a ** nontrivial counterexample** of a positive real $a$ and a sequence $s_n$ such that the fractional part of the sequence $as_n$ is equidistributed modulo 1 but the sequence of the ratios $s_i/s_n$ do not approach equidistribution modulo 1 as $n \to \infty$.

**Edit**: I am explaining what I mean by *nontrivial* because the example given by Noam indicates that it is necessary to explain it explicitly. The examples of Weyl, Vinogradov and Hlawka are nontrivial because there is no assumption on the normality constant. If we take the constant to be normal, we indirectly already assume what we want to prove and so we can construct many artificial examples.