# Distribution of $\{n^p\alpha\}$ for irrational $\alpha$

Let $$\alpha$$ be an irational number. Consider sequence $$x_n=\{n^p\alpha\}$$, $$n\in\mathbb{N}$$ (it's the fractional part of $$n^p\alpha$$), where $$p$$ is a nonzero real number.

Question. For which values of $$p$$ the sequence $$\{x_n\}_{n\in\mathbb{N}}$$ is equidistributed on $$[0,1)$$? The other qusetion is when $$\{x_n\}_{n\in\mathbb{N}}$$ is dense on $$[0,1)$$.

It's known that if $$p\in\mathbb{N}$$, then it's true (it's a consequence of Weyl's equdsitribution criterion and van der Corput's difference theorem). However, it's not clear how to apply Weyl's criterion in case when $$p\notin\mathbb{N}$$. I encountered similar problem when I was working on this question Convergence of the product $\prod_{n=1}^{\infty}\left(1+\frac{x^n}{n^p}\right)\cos\frac{x^n}{n^q}$ (in order to study the behaviour of $$\cos\frac{1}{n^q}$$, so $$\alpha=1/\pi$$ in this case).

Update. It would be also intersting to investigate the distbution of $$\left\{\frac{x^n}{n^p}\right\}$$.

If the result is known, please give a link or reference. Any help would be appreciated.

• One can, I believe, get an (oscillating) asymptotic formula for $\sum x_n$. I recommend checking Montgomery's Ten Lectures on the Interface between Analytic Number Theory and Harmonic Analysis. May 27, 2020 at 17:34

Let $$(f(n))$$, $$n=1,2,\ldots,$$ be a sequence of real numbers, and let $$k$$ be a positive integer. If $$\Delta^k f(n)$$ is monotone in $$n$$, if $$\Delta^k f(n) \rightarrow 0$$ and $$n|\Delta^kf(n)|\rightarrow\infty$$ as $$n\rightarrow\infty$$, then the sequence $$(f(n))$$ is uniformly distributed modulo $$1$$.
Here $$\Delta f(n)=f(n+1)-f(n)$$ and $$\Delta^k = \Delta \circ \Delta^{k-1}$$. When $$p>0$$ is real and not an integer, then we can find $$k$$ that satisfies conditions of the Theorem. Further, it is easy to see that $$p\leq 0$$ does not give uniform distribution. Hence, your sequence is uniformly distributed if and only if $$p>0$$.
• Thanks. Am I right that we should choose $k=[p]$? May 28, 2020 at 7:28
• Actually $\lfloor p \rfloor +1$. May 28, 2020 at 16:11
• By the way, it's easier to apply here the next theorem (3.5) (just the same but in differential form, i. e. if $f^{(k)}(x)\to 0$ monotonically and $x|f^{(k)}(x)|\to+\infty$, when $x\to\infty$, then we have uniform distribution of $(f(n))$). Dec 1, 2020 at 8:00