I think that there is a typo in the paper referred to in the title of the question, available here. While discussing Roth's proof of AP discrepancy, on page 2, the author states that

Roth [20] established the existence of such irregularities of distribution. Indeed, more generally he showed that if $A$ is a subset of the [positive] integers up to $N$ with $|A| = ρN$, then there exists an absolute positive constant $c$ such that $$ \sup_{\stackrel{k \leq \sqrt{N}}{a\pmod k}} \left| \sum_{\stackrel{n \in A}{n \equiv a \pmod k}} 1-\frac{\rho N}{q} \right| \geq c \sqrt{\rho(1-\rho)}N^{1/4} $$ In other words, the only subsets of $[1, N ]$ that are evenly distributed in all arithmetic progressions with moduli below $\sqrt{N}$ are essentially the empty set (with $\rho=1$) or the whole set (with $\rho=1$).

Shouldn't the $k$'s in this inequality should be replaced by $q$'s?