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}ψ_3({3^j x}), ] where [ ψ3(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 $ω_3$; for example, [ T_3(1/3)=4/9, \qquad ω_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 ω_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 $0<α<1$, we prove the endpoint estimate [ |S_1(\mathbbm 1_A)|α\ge |A|^*, ] and show that it is sharp up to a constant depending only on $α$ and $n$.

• The paper establishes a sharp isoperimetric profile for ternary martingale filtrations using an explicit Bellman function that outperforms classical Takagi-type functions.
• It introduces n-adic lower bounds derived via digit-sum Bellman functions, achieving optimal logarithmic order for boundary energy minimization.
• The research extends discrete isoperimetry to recursive n-adic structures, linking analytic martingale theory with number-theoretic digit decompositions.

Sharp Martingale Isoperimetry for Ternary and $n$-adic Filtrations

Introduction and Motivation

The paper "Sharp Ternary Martingale Isoperimetry and $n$-adic Takagi-Type Lower Bounds" (2607.11069) addresses martingale isoperimetric profiles associated with regular $n$-adic filtrations on $[0,1)$. The central object is the $L^1$ martingale variation of indicator functions, interpreted as a boundary energy on rooted regular $n$-ary trees. The connections to discrete isoperimetric inequalities, digit-sum statistics, and generalized Takagi-type functions (including the Takagi--van der Waerden function) are explicitly developed, with precise results in the ternary ($n=3$) case and robust lower bounds in general bases.

The work is situated at the intersection of analytic martingale theory, combinatorial isoperimetry, and number-theoretic digit decomposition. Its methods extend boundary minimization problems from the classical Boolean cube to more general recursive structures, where the underlying filtration is $n$-adic.

Main Results and Claims

Exact Sharp Profile for Ternary Filtration

A main result is the explicit solution of the isoperimetric profile for the ternary ($n=3$) martingale filtration:

$$V_3(x) = T_3(x) = \sum_{j=0}^\infty 3^{-j} \psi_3(\{3^j x\}),$$

where $n$0 is a piecewise-linear generator reflecting mean absolute deviation (not simply range), and $n$1 denotes the fractional part. Notably, $n$2 is not the usual ternary Takagi--van der Waerden function $n$3, so at points such as $n$4, $n$5 while $n$6—a direct refutation of naive analogies or generalizations. The authors rigorously establish that $n$7 is the Bellman function maximizing the profile for ternary isoperimetry.

Figure 1: Comparison of Takagi-type candidates on $n$8-adic grids; for $n$9, $n$0 exceeds $n$1 at specific points, illustrating non-equivalence of classical and sharp Bellman functions.

$n$2-adic Lower Bounds via Digit-Sum Bellman Functions

For general $n$3, the paper shows that every measurable set $n$4 satisfies

$n$5

and proves that this logarithmic order is sharp up to a constant. The main methodological innovation involves constructing a valid Bellman function $n$6 using base-$n$7 digit-sum statistics, which admits the Takagi-type series expansion:

$n$8

with a generator $n$9 having strictly stronger pointwise lower bounds than $[0,1)$0 for $[0,1)$1 (see Figure 1). For dyadic ($[0,1)$2) and ternary ($[0,1)$3), $[0,1)$4 holds, but for $[0,1)$5 the Bellman function $[0,1)$6 strictly exceeds $[0,1)$7 except at special rationals.

Endpoint Behavior and Quasi-Norm Sharpness

For $[0,1)$8, the paper establishes that the boundary-distance function $[0,1)$9 is itself admissible for the Bellman inequality, yielding

$L^1$0

with optimality up to constants, as shown by explicit sequences of $L^1$1-adic intervals.

Methodological Details and Innovations

Recursive Bellman Principle

The Bellman admissibility principle is generalized to $L^1$2-point mean absolute deviation, not just maximum/minimum splitting, and reduced to a recursive inequality involving piecewise-linear generators. Formal verification exploits quantifier elimination and exact computation on finite cells.

Ternary Compression and Debt Propagation

The sharpness verification for $L^1$3 uses a rigorous induction enhanced with a local "debt" function $L^1$4, allowing closure of intricate recursive inequalities through computer-assisted symbolic quantifier elimination. The explicit splitting structure, with at most one residual child at each ternary step, is critical for achieving sharpness.

Digit-Sum Inequalities and Number-Theoretic Bellman Construction

The authors leverage recent summatory digit-sum inequalities to construct the Bellman function $L^1$5 for general $L^1$6. This approach relates isoperimetric optimization to digit decompositions, echoing Hart's formula and connecting to digit-sum and Takagi-type function theory in fractal analysis and combinatorics.

Asymptotics and Contradictory Claims

A central claim contradicted in the paper is that the ternary isoperimetric profile aligns with the Takagi--van der Waerden function. Instead, the profile is strictly larger for specific points, and the naive extension of the one-residual generator to $L^1$7 fails Bellman admissibility, as shown both analytically and computationally.

Numerically, the asymptotics of $L^1$8 and $L^1$9 are shown to be $n$0 for $n$1, establishing not only lower bounds but also sharpness up to constants. The lower bounds for $n$2-adic intervals $n$3 are attained up to the explicit constant $n$4.

Implications and Future Directions

The results have direct implications for sharp isoperimetric inequalities in product structures beyond the Boolean cube, providing precise boundary profiles for martingale filtrations. Practically, these bounds inform discrete harmonic analysis, Boolean function theory, and optimization in recursive graph models.

Theoretically, the paper opens new questions regarding the explicit structure of sharp $n$5-adic Bellman functions for $n$6, where classical Takagi-type or one-residual approaches fail. Future advances may involve combinatorial or analytic characterization of admissible generators, or may reveal deeper number-theoretic structures governing recursive boundary energies.

The methodology sets a standard for combining analytic recursion, computer-assisted quantifier elimination, and digit-sum statistics in constructive Bellman proofs. Speculatively, connections to self-similar, fractal, and automatic sequence analysis will continue to inform optimal boundary behaviors in both finite and infinite recursive structures.

Conclusion

This paper establishes the exact ternary martingale isoperimetric profile, demonstrates that Takagi-type functions must be adapted for $n$7-adic settings, constructs robust $n$8-adic lower bounds via digit-sum Bellman functions, and proves sharp logarithmic asymptotics. The results reject simplistic generalizations and link analytic isoperimetry to number-theoretic digit decompositions. The techniques and findings suggest new directions for rigorous optimal boundary-energy analysis in recursive filtrations, martingale theory, and digital combinatorics.