Growth bound for signs of m-term linear combinations
= Growth bound for signs of m-term linear combinations
For $f=\sum_{j=1}^m\alpha_jg_j$ with $g_j$ drawn from a binary class $\mathcal G$, fixing $(g_1,\ldots,g_m)$ leaves an $m$-dimensional linear sign class. Consequently
$$
s(\operatorname{sgn}\mathcal F,n)
\leq(n+1)^m|\mathcal G|^m
$$
when $\mathcal G$ is finite, and in general
$$
s(\operatorname{sgn}\mathcal F,n)
\leq(n+1)^m s(\mathcal G,n)^m.
$$