Critical Weierstrass modulus ($\omega (g_{5,5},\delta )\asymp \delta \log (1/\delta )$)

For $g(x)={\sum }_{k\ge 0}{5}^{-k}\cos (5^{k}x)$, all frequencies satisfy $5^k\equiv 1(\mathrm{mod}4)$. At $x=\pi /2$, the increment is $-{\sum }_{k\ge 0}{5}^{-k}\sin (5^{k}h)$. If $5^m\le 1/h<5^{m+1}$, the first $m+1$ terms have $0<5^{k}h\le 1$ and contribute at least $(m+1)h\sin 1$, while the remaining tail has magnitude at most $5^{-m}/4\le 5h/4$. 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 $E_n(g)\asymp n^{-1}$.