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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

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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 $\{1,2,\ldots,n\}.$

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