For any $C>$ 0 in any infinite $\pm 1$ sequence $\left(x_n\right)$

there exists a subsequence $x_d, x_{2 d}, x_{3 d}, \ldots, x_{k d}$,

for some positive integers $k$ and $d$,

such that $\left|\sum_{i=1}^k x_{i d}\right|>C$. 

Erdős Discrepancy Conjecture

$\begin{aligned} \phi_{(l, C, d)}=s_0^{(1, d)} \bigwedge_{i=1}^{\left\lfloor\frac{l}{d}\right\rfloor}[ & \bigwedge_{-C \leq j<C}\left(s_j^{(i, d)} \wedge p_{i d} \rightarrow s_{j+1}^{(i+1, d)}\right) \wedge \\ & \bigwedge_{-C<j \leq C}\left(s_j^{(i, d)} \wedge \neg p_{i d} \rightarrow s_{j-1}^{(i+1, d)}\right) \wedge \\ & \left(s_C^{(i, d)} \wedge p_{i d} \rightarrow B\right) \wedge \\ & \left.\left(s_{-C}^{(i, d)} \wedge \neg p_{i d} \rightarrow B\right)\right]\end{aligned}$

$\phi_{(l, C)}=\neg B \wedge \bigwedge_{d=1}^{\left\lfloor\frac{l}{C+1}\right\rfloor} \phi_{(l, C, d)} \wedge$ frame $_{(l, C)}$,

where frame $(l, C)$ is a Boolean formula encoding that the automaton state is correctly defined,

that is,

exactly one proposition from each of the sets $\left\{s_j^{(i, d)} \mid-C \leq j \leq C\right\}$, for $d=1, \ldots,\left\lfloor\frac{l}{C+1}\right\rfloor$ and $1 \leq i \leq\left\lfloor\frac{l}{d}\right\rfloor$, is true in every model of $\phi_{(l, C)}$.

The formula $\phi_{(l, C)}$ is satisfiable if, and only if, there exists $a \pm 1$ sequence $\bar{x}=x_1, \ldots, x_l$ of length $l$ of discrepancy $C$.

Moreover, if $\phi_{(l, C)}$ is satisfiable, the sequence $\bar{x}=x_1, \ldots, x_l$ of discrepancy $C$ is uniquely identified by the assignment of truth values to propositions $p_1, \ldots p_l$.