# Conjecture about an Exponential Sum

Let $$X \subset \mathbb{N}$$ and say that $$X$$ is super-equidistributed if for all $$\alpha \in \mathbb{R} \setminus \mathbb{Z}$$ there exists $$C(\alpha) > 0$$ such that for all $$N$$ $$\left| \sum_{x \in X, x \leq N} \exp(2\pi i \alpha x) \right| < C(\alpha).$$

My conjecture is the following: If $$X$$ is super-equidistributed, then either $$X$$ is finite or $$\mathbb{N} \setminus X$$ is finite.

Why should this be true? Well there isn't too much evidence but one can show that if $$X$$ is super-equidistributed, then for all integers $$a,q$$ we have $$\#\{x \in X: x \leq N, x \equiv a \pmod{q}\} = \frac{|X \cap [1,N]|}{q} + O_q(1).$$

This is a strong condition on $$X$$, but it is not enough. For example a friend of mine suggested the set $$S = \bigcup_{n \geq 1} [n!, 2 \cdot n!) \cap \mathbb{Z}$$. This satisfies this property mod $$q$$ for all $$q$$ but is neither finite nor is $$\mathbb{N} \setminus S$$. However if we pick $$\alpha = e$$, one can show that $$\sum_{x \in S, x \leq N} \exp(2 \pi i \alpha x)$$ is unbounded. To see this one essentially uses that the fractional part of $$e \cdot k!$$ is very well behaved for any $$k$$.

Also note that any set $$X$$ such that $$X$$ or $$\mathbb{N} \setminus X$$ is finite is super-equidistributed since $$\sum_{x \geq 0} \exp(2 \pi i \alpha x) = (1 - \exp(2 \pi i \alpha))^{-1}$$.

I would appreciate any thoughts on this problem. Possible counter-examples, solutions or consequences are all welcome.

• Of possible interest is W Narkiewicz, Uniform distribution of sequences of integers in residue classes, in the Springer Lecture Notes series. – Gerry Myerson Jan 10 at 0:36
• On a first glance it seems there are results about when a multiplicative sequence is uniformly distributed. Maybe one can try to establish the result for X multiplicative and somehow reduce to this case? – 1213 Jan 10 at 0:51
• One could consider also a version where we replace $X$ with a $\pm 1$-valued sequence $x_n$ and demand $\sum_{n=1}^N x_n \exp(2 \pi i \alpha n)$ bounded for all $\alpha$, even zero. – Will Sawin Jan 10 at 1:15
• Does there exist such a $\pm 1$ sequence? The erdos discrepancy problem tells us that there doesn't exists such a sequence for which $\sum_{n=1}^N x_n \exp(2\pi i \alpha n)$ is bounded independently of $\alpha$. – 1213 Jan 10 at 1:26
• Does the Thue-Morse sequence (or rather, $X$ being the set of indices of $1$s in the Thue-Morse sequence) provide a counterexample? It's similar to the $mod 2^n$ set, but seems like it should work for dyadic roots of unity. – user44191 Jan 10 at 1:48

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.