I have a conjecture that is stronger than the Erdos discrepancy conjecture, can someone think of a counter example?
Let $S$ be any large set and let $(x_1,x_2,...)$ be any infinite $\pm 1$ sequence, then for any integer $C$, we can find an integer $k$ and an $s \in S $ so that $\sum_{i=1}^k|x_{i \cdot s}| > C$
I also think that the inverse is true. Thus, if $S$ is small there exists an integer $C$ and some $(x_1,x_2,...)$ so for every $s$ and $k$ we have that $\sum_{i=1}^k|x_{i \cdot s}| <C$
