# Discrepancy in non-homogeneous arithmetic progressions

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

The discrepancy for arbitrary Arithmetic Progessions over the natural numbers (of which Homogeneous APs are a subset) was indeed shown by Roth to be infinite. But he showed more, with a lower bound of $$c\cdot n^{1/4}$$ for the progressions restricted to the set $$\{1,2,\ldots,n\}$$. This bound was later proved to be essentially tight.
Interestingly, Lovasz used Semidefinite programming to derive the result with $$c=1/49.$$
Tao's result shows that the discrepancy restricted to homogeneous AP's is also infinite. However, as of now, there is no effective lower bound on the discrepancy of homogeneous APs restricted to $$\{1,2,\ldots,n\}.$$