Is this a criterion for uniform distribution modulo one?

Question

For all $k \in \mathbb N$ let $a_k$ be strictly positive bounded weights, i.e. there are constants $C_1$ and $C_2$ such that $0<C_1\le a_k \le C_2$. Now a real valued sequence $(x_k)_{k \in \mathbb N}$ satisfies $\lim_{n \to \infty}\frac{1}{n} \sum_{k=1}^n (a_k)^m e^{2\pi i x_km}=0$ for all non-zero integers $m$. Does this imply that the sequence $x_k$ is uniformly distributed modulo one?

It is similar to the Weyl criterion and would surely not be true if we consider just weights $a_k$ where the condition would only imply weighted uniform distribution modulo one. But with the weights $(a_k)^m$ depending on the parameter $m$ this could be true.

Any thoughts or potential counterexamples would be highly appreciated.

Answer 1

I provide a sequence $(x_k)_k$ that is not uniformly distributed but satisfies $\frac{1}{N}\sum_{k \le N} a_k^m e^{2\pi i x_k m} \to 0$ as $N \to \infty$ for each $m \ge 1$, where $a_k = 1$ if $k$ is odd and $a_k = 3$ if $k$ is even. Let $x_k = 0$ if $k$ is odd. Then, for any $m \ge 1$, $\sum_{\substack{k \le N \\ 2 \nmid k}} e^{2\pi i x_k m} = \frac{1}{2}N+o(N)$, so it suffices to have, for any $m \ge 1$, $\sum_{\substack{k \le N \\ 2 \mid k}} e^{2\pi i x_k m} = \frac{-1}{2\cdot 3^m}N+o(N)$.

Rescaling, our general problem is to find a sequence $(x_k)_k$ such that $$\sum_{k \le N} e^{2\pi i x_k m} = \frac{-1}{3^m}N+o(N)$$ for each $m \ge 1$. I feel like there's a reference that guarantees this result, but let me say what I came up with (which has probably already been done somewhere). Let $f : \mathbb{T} \to \mathbb{C}$ be such that $\hat{f}(0) = 1$, $\hat{f}(m) = -\frac{1}{3^m}$ for $m \ge 1$, and $\hat{f}(m) = \overline{\hat{f}(-m)}$ for $m \le -1$. Since the fourier coefficients decay rapidly, $f$ is continuous. Also, $f$ is real, due to how the negative fourier coefficients are defined. In fact, $f(x) = 1-2\sum_{m \ge 1} \frac{1}{3^m}\sin(2\pi m x)$, which in particular shows $f$ is non-negative. Therefore, for each $N \ge 1$, we may take rational approximations $\frac{j_0}{M_N},\dots,\frac{j_{N-1}}{M_N}$ to $f(\frac{0}{N}),f(\frac{1}{N}),\dots,f(\frac{N-1}{N})$ with $j_i \in \mathbb{Z}^{\ge 0}$ for each $i$. Let $s_N$ denote a block of length $T_N := j_0+\dots+j_{N-1}$ that has $\frac{t}{N}$ appearing $j_t$ times. Define our sequence $(x_k)_k$ by $2^{2^1+T_1}$ consecutive $s_1$'s, followed by $2^{2^2+T_2}$ consecutive $s_2$'s, followed by $2^{2^3+T_3}$ consecutive $s_3$'s, etc.. Then, for any $m \ge 1$, if $N$ is large enough, $$\frac{1}{N}\sum_{k \le N} e^{2\pi i x_k m} \approx \frac{1}{N_0}\sum_{t=0}^{N_0-1} f(\frac{t}{N_0})e^{2\pi i \frac{t}{N_0}m} \approx \int_0^1 f(x)e^{2\pi i xm}dx = \frac{-1}{3^m},$$ for some large $N_0$, the first approximation holding due to $f$ having mean $1$ (I'll leave details/quantifiers to you).