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. 



Is this a criterion for uniform distribution modulo one?
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). 


