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

Abstract

Let $S_1$ be the one-variation associated with the regular $n$-adic martingale filtration on $[0,1)$. We study the martingale isoperimetric profile \[ V_n(x):= \inf_{\substack{A\subset[0,1)\ {\rm measurable}\\ |A|=x}} \|S_1(\mathbbm 1_A)\|_1 . \] For the ternary filtration we determine this profile exactly. Namely, \[ V_3(x)=T_3(x):= \sum_{j=0}^{\infty}3^{-j}\psi_3(\{3^j x\}), \] where \[ \psi_3(t)= \min\left\{ \frac{1+2\left|t-\frac12\right|}{3}, \frac{2-4\left|t-\frac12\right|}{3} \right\}, \qquad 0\le t\le1 . \] 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, \[ T_3(1/3)=4/9, \qquad \omega_3(1/3)=1/3 . \] For general $n\ge2$, we prove that every measurable $A\subset[0,1)$ satisfies \[ \|S_1(\mathbbm 1_A)\|_1 \ge \omega_n(|A|^*) \asymp_n |A|^*\log\frac1{|A|^*}, \qquad |A|^*:=\min\{|A|,1-|A|\}. \] 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 \[ \|S_1(\mathbbm 1_A)\|_\alpha\ge |A|^*, \] and show that it is sharp up to a constant depending only on $\alpha$ and $n$.