Sharp Ternary Martingale Isoperimetry and $n$-adic Takagi-Type Lower Bounds

Let $S_1$ be the one-variation associated with the regular $n$-adic martingale filtration on $[0,1)$. We study the martingale isoperimetric profile $$V_n(x):=\underset{\begin{array}{c}A\subset [0,1)\mathrm{m}\mathrm{e}\mathrm{a}\mathrm{s}\mathrm{u}\mathrm{r}\mathrm{a}\mathrm{b}\mathrm{l}\mathrm{e}\\ \mathrm{\mid }A\mathrm{\mid }=x\end{array}}{\inf }\mathrm{\parallel }{S}_1(\text{mathbbm}{1}_A){\mathrm{\parallel }}_1\mathrm{.}$$ For the ternary filtration we determine this profile exactly. Namely, $$V_3(x)=T_3(x):=\sum _{j=0}^{\mathrm{\infty }}{3}^{-j}{\psi }_3(\{3^{j}x\}),$$ where $${\psi }_3(t)=\min \{\frac{1+2\mid t-\frac{1}{2}\mid }{3},\frac{2-4\mid t-\frac{1}{2}\mid }{3}\},0\le t\le 1.$$ Thus the sharp ternary profile is a Takagi-type Bellman function. It is, however, not the usual ternary Takagi–van der Waerden function ${\omega }_3$; for example, $$T_3(1\mathrm{/}3)=4\mathrm{/}9,{\omega }_3(1\mathrm{/}3)=1\mathrm{/}3.$$ For general $n\ge 2$, we prove that every measurable $A\subset [0,1)$ satisfies $$\mathrm{\parallel }{S}_1(\text{mathbbm}{1}_A){\mathrm{\parallel }}_1\ge {\omega }_n(\mathrm{\mid }A{\mathrm{\mid }}^{\ast }){\asymp }_n\mathrm{\mid }A{\mathrm{\mid }}^{\ast }\log \frac{1}{\mathrm{\mid }A{\mathrm{\mid }}^{\ast }},\mathrm{\mid }A{\mathrm{\mid }}^{\ast }:=\min \{\mathrm{\mid }A\mathrm{\mid },1-\mathrm{\mid }A\mathrm{\mid }\}\mathrm{.}$$ Moreover, this logarithmic order is sharp up to a constant depending only on $n$. Finally, for every $0<\alpha <1$, we prove the endpoint estimate $$\mathrm{\parallel }{S}_1(\text{mathbbm}{1}_A){\mathrm{\parallel }}_{\alpha }\ge \mathrm{\mid }A{\mathrm{\mid }}^{\ast },$$ and show that it is sharp up to a constant depending only on $\alpha$ and $n$.