Conjecture about an exponential sum

This was proven by Reynold Fregoli! https://arxiv.org/abs/1912.08626

Not quite an answer, but maybe it points to one:

Ivan Niven, Uniform distribution of sequences of integers, Compositio Mathematica 16 (1964) 158-160, defines a sequence $A=(a_1,a_2,\dots)$ of integers to be uniformly distributed if $$A(n,j,m)={n\over m}+{\rm o}(n)$$ for every integer $m\ge2$ and every $j$, $1\le j\le m$, where $A(n,j,m)$ is the number of terms among $a_1,a_2,\dots,a_n$ satisfying $a_i\equiv j\bmod m$. He gives as an example the sequence $[\theta],[2\theta],\dots$ of integer parts of the multiples of $\theta$, where $\theta$ is any real irrational. He cites a result of Uchiyama to the effect that $A$ is uniformly distributed if and only if $$\sum_{k=1}^Ne^{2\pi iha_k/m}={\rm o}(N)$$ for all positive integers $m$ and $h$, $1\le h\le m-1$. He gives some applications.

The Uchiyama reference is On the uniform distribution of sequences of integers, Proc Jap Acad 37 (1961) 605-609. There is also an earlier paper of Niven on the topic, Uniform distribution of sequences of integers, Trans Amer Math Soc 98 (1961) 52-61.