**O(1) or o(logn) discrepancy for multiples of an irrational for at least one sub interval.**


Using $\{x\}$ to denote the fraction part of $x$ we can define for any $I\subset [0,1]$,


$$E(n,\theta, I) ={ \left|\{\,\{\theta\},\{2\theta\},\dots,\{n\theta\} \,\} \cap I \right|}-n|I|$$
$ $
$$\Delta\_{sup}(n,\theta)=\sup\_I |E(n,\theta,I)|$$. 


The [Equidistribution Theorem](https://en.wikipedia.org/wiki/Equidistribution_theorem) says that the sequence $a\_i=(i\theta)$, $i\in\mathbb{Z}\_{\geq1}$ is [equidistributed modulo 1](https://en.wikipedia.org/wiki/Equidistributed_sequence#Equidistribution_modulo_1) when $\theta$ is irrational hence


$$\Delta\_{sup}(n,\theta)=o(n)$$ for all irrational $\theta$.


Furthermore if $\theta$ has bounded partial denominators in its continued fraction expansion then we have 


$$\Delta\_{sup}(n,\theta)\ll \log n$$ 


(See Theorem 1.B of W Schmidt "Lectures on irregularities of distribution" $^\*$


My question asks how small can $E$ be as a function of $n$. 


**1) Does there exist an irrational number $\theta$ and a fixed interval $I\subsetneq [0,1]$ with $E(n,\theta, I)=O(1)$?**


If this false or too difficult:


**2) Does there exist an irrational number $\theta$ and a fixed interval $I\subsetneq [0,1]$ with $E(n,\theta, I)=o(\log n)$?**


If this false or too difficult:


**3) Does there exist an irrational number $\theta$ and an interval $I\subsetneq [0,1]$ of fixed size whose position is allowed to vary with $n$, s.t. $E(n,\theta, I)=o(\log n)$?**


$\*$ "*Schmidt, Wolfang M.*, Lectures on irregularities of distribution. (Notes by T. N. Shorey), Tata Institute of Fundamental Research, Lectures on Mathematics and Physics: Mathematics, 56. Bombay: Tata Institute of Fundamental Research. vi, 128 p. (1977). [ZBL0434.10031](https://zbmath.org/?q=an:0434.10031).")



O(1) or o(logn) discrepancy for multiples of an irrational for at least one sub interval
I think you are asking for so-called "bounded remainder sets". Given $\theta$, there exist intervals $I$ having discrepancy $O(1)$, namely those whose length is in $\mathbb{Z} + \theta \mathbb{Z}$. The classical reference is a paper of Kesten: 


H. Kesten,On a conjecture of Erdös and Szüsz related to uniform distribution mod 1, Acta Arith. 12(1966), 193–212.


