= Critical Weierstrass modulus
{title2=$\omega(g_{5,5},\delta)\asymp\delta\log(1/\delta)$}
For $g(x)=\sum_{k\ge0}5^{-k}\cos(5^kx)$, all frequencies satisfy $5^k\equiv1\pmod4$. At $x=\pi/2$, the increment is $-\sum_{k\ge0}5^{-k}\sin(5^kh)$. If $5^m\le1/h<5^{m+1}$, the first $m+1$ terms have $0<5^kh\le1$ and contribute at least $(m+1)h\sin1$, while the remaining tail has magnitude at most $5^{-m}/4\le5h/4$. This proves a lower <modulus of continuity> bound of order $h\log(1/h)$ for small $h$. The <inverse theorem for trigonometric approximation> supplies the matching upper bound because the <positive lacunary trigonometric series> has $E_n(g)\asymp n^{-1}$.
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