I have a doubt, Roth's discrepancy theorem [1] says that there is a subset of arithmetic progressions $A \in [n]$ where any function $f:N\rightarrow \left \{ -1,1 \right \} $ implies

$ \left | \sum _{a\in A}f(a) \right |\geq cn^{1/4} $

(Does not specify if the arithmetic progression is homogeneous or not homogeneous)

In 2015 Terence Tao [2] proved for any homogeneous arithmetic progression we have

$Sup_{n,d\in N}\left | \sum^{n} _{j=1}f(jd) \right |=\infty$

It occurred to me to ask if the discrepancy is infinite for non-homogeneous arithmetic progressions or is it an open question

[1]Roth, K. F. Remark concerning integer sequences. Acta Arith. 9 1964 257–260.

[2] Tao, Terence (2016). "The Erdős discrepancy problem". Discrete Analysis: 1–29. arXiv:1509.05363