TITLE: For $x$ irrational, is $a_{n} =\sum_{k=1}^{n}(-1)^{⌊kx⌋}$ unbounded?
QUESTION [26 upvotes]: For $x$ irrational, define $a_{n} :=\sum_{k=1}^{n}(-1)^{⌊kx⌋}$. Can you prove that $\left\{a_n\right\}$ is unbounded?
I feel that it is not easy to treat every irrational $x$.
I have asked in S.E. and it seems that it is an advanced question. Thus I go for help here.

REPLY [4 votes]: As Ville Salo already wrote in his answer, your question can be phrased in terms of the difference between the number of elements of the sequence $y, 2y, \dots, ky$ which are contained in $[0,1/2]$, and $k$ times the length of $[0,1/2]$. Here $y = x/2$. In the language of discrepancy theory, you are asking whether the interval $[0,1/2]$ is a so-called bounded remainder set of the sequence $(n y~\text{mod}~1)_{n \geq 1}$. However, it is known that it is not: the only intervals with bounded remainder are those whose length is in $\mathbb{Z} + y \mathbb{Z}$, and since $y$ is irrational in your question the length of $[0,1/2]$ is not of such type. Bounded remainder sets were classified in this paper: H. Kesten, On a conjecture of Erdös and Szüsz related to uniform distribution mod 1, Acta Arith. 12(1966), 193–212.