**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 says that the sequence $a_i=(i\theta)$, $i\in\mathbb{Z}_{\geq1}$ is equidistributed 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.")