Large discrepancy in homogeneous quasi-arithmetic progressions (Q2495694)

The set of natural numbers \(\{0, [\alpha], [2\alpha],\dots, [m\alpha]\}\), where \(m\) is a natural number, \(\alpha\) a real number and \([\,]\) is the greatest integer function, is called a quasi-arithmetic progression. The number \(\alpha\) is called the common difference. Let \(\mathbb N\) be the set of all natural numbers. Main Theorem: Given any function \(f: \mathbb N \rightarrow \{-1, +1 \},\) and any integer \(t \geq 1\), then for all sufficiently large \(n\), there is some homogeneous quasi-arithmetic progression \(A\) with common difference between \(t\) and \(t+1\), and largest term less than \(n\), such that \[ \left|\sum_{a \in A} f(a)\right| > \frac{1}{24\sqrt{t}}\cdot(\log_2 n)^{1/4}. \]