Let $\mathbb{F}$ be a finite prime field of characteristic different than $2$ and $\beta \in \mathbb{F}$ a generator of the $2$-power order multiplicative subgroup of order $2^k$, so $\beta^{2^{k-1}} = -1$. Consider sums of the form $\sigma(a) = \sum_{i = 0}^{2^{k-1} - 1} a_i \beta^i$ with $a_i \in \{\pm 1\}$. Can we say anything about the maximum, over all $\sigma$, of the number of sequences $a = (a_i)_{i=0}^{2^{k-1} - 1}$ such that $\sigma(a) = \sigma$?
We can phrase this a bit differently in terms of subset sums: the condition $\sigma(a) = \sigma(b)$ is equivalent to requiring $\sum_{i=0}^{2^{k-1} - 1} (a_i - b_i) \beta^i = 0$, where now $a_i - b_i \in \{0, \pm 2\}$. Of course we can divide by $2$, so we're looking to count the number of $c_i \in \{0, \pm 1\}$ such that $\sum_{i=0}^{2^{k-1} - 1} c_i \beta^i = 0$.
Now, let $S_+$ be the subset of $\{0, \ldots, 2^{k-1} -1\}$ with $c_i = +1$ and $S_-$ the subset with $c_i = -1$. Since $\beta^{2^{k - 1}} = -1$, we can rewrite this as $\sum_{i \in S_+} \beta^i + \sum_{i \in S_-} \beta^{2^{k-1} +i} = 0$. So we're counting the number of subsets $S \subseteq \{0, \ldots, 2^k -1\}$ such that $\sum_{i \in S} \beta^i = 0$ with the condition that $S$ contains at most one of $\{i, 2^{k-1} + i\}$ for each $i \in \{0, \ldots, 2^{k-1} -1\}$.
This seems close to the sort of "zero sum sequence" problem that I know has received some extensive attention in the (additive?) combinatorics literature, but my knowledge of said literature is quite superficial, so I'm hopeful that there are some known techniques to address this sort of question!
