Assume that $1<a<2$ and $c\ne 0$ is a real number. What is known about the distribution of the sequence $cn^a$ modulo 1? Say, is it true that for certain $\theta<1$ (depending on $a$ and $c$) we have $|\sum_{k=1}^n \{ck^a\}-\frac{n}2|=O(n^\theta)$?
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asked Nov 20, 2018 at 19:32
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$\begingroup$ K does not appear in your sum except as an index. Is this intentional? Gerhard "Smells Typo In The Air" Paseman, 2018.11.20. $\endgroup$– Gerhard PasemanCommented Nov 20, 2018 at 20:11
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$\begingroup$ For the exponential sum $\sum e^{2\pi i ck^a}$ one may try to use Weyl differencing so that $(k+1)^a - k^a \sim a k^{a-1}$ appears in the exponent. If this gives a non-trivial upper bound then the result you want might depend on the irrationality measure of $c$. See mathoverflow.net/questions/35902/… and math.stackexchange.com/questions/2270/… for some details. $\endgroup$– François BrunaultCommented Nov 20, 2018 at 21:43
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1 Answer
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Exercise 2.23 in Kuipers and Niederreiter, Uniform Distribution of Sequences, goes: Use Theorem 2.7 to show that the sequence $\{\alpha n^{\sigma}\}$, $n=1,2,\dots$, $\alpha\ne0$, $1<\sigma<2$, is uniformly distributed modulo one.
Theorem 2.7 is as follows. Let $a$ and $b$ be integers with $a<b$, and let $f$ be twice differentiable on $[a,b]$ with $f''(x)\ge\rho>0$ or $f''(x)\le-\rho<0$ for $x\in[a,b]$. Then $$\left|\sum_{n=a}^be^{2\pi if(n)}\right|\le(|f'(b)-f'(a)|+2)\left({4\over\sqrt{\rho}}+3\right)$$
answered Nov 20, 2018 at 22:07
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1$\begingroup$ Oh, almost what I need. Is there a standard way to come from the estimates of exponential sums to the estimates of functions like $\{x\}-1/2$? We may try to expand it in Fourier series and interchange series and a sum and use this estimate, but maybe there are more clever general ways? $\endgroup$ Commented Nov 20, 2018 at 22:16
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$\begingroup$ as Dmitry Krachun suggests, the appropriate general way to try is en.m.wikipedia.org/wiki/… it remains to substitute and check whether anything is ok $\endgroup$ Commented Nov 20, 2018 at 22:33
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3$\begingroup$ The standard way to come from exponential sums to such estimates is as follows: a) exponential sums can be used to bound something called "discrepancy", but the Erdös-Turan inequality. b) the discrepancy can be used to estimate the sort of sums you have, by Koksma's inequality (deviation is bounded by variation of the function, multiplied with discrepancy). The standard reference is the book of Kuipers-Niederreiter, which has all these things. $\endgroup$ Commented Nov 21, 2018 at 8:33