CF 102992H - Harmonious Rectangle

CF 102992H - Harmonious Rectangle

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Solution

Solution

Let $T=\binom{2t-1}{t}$ and write $x=N/T$. In Section 7.2.1.3, the number $\kappa_{tN}$ is expressed through the binary representation of $N$ by decomposing the corresponding $(s,t)$-combination into the associated compositions $q_t,\dots,q_0$ of (11). The argument leading to Exercise 84 shows that the fluctuation of $\kappa_{tN}-N$ is governed by the same additive structure that defines the Takagi function, namely the dyadic contributions of the form $r_k(x)=(-1)^{\lfloor 2^k x\rfloor}$ accumulated through the binary carry structure of the representation (14).

The functions $\lambda_{tN}$ and $\mu_{tN}$ arise from the same decomposition but with extremal choices of the intermediate binary configurations consistent with a fixed prefix of the $(s,t)$-representation of $N$. Equivalently, when one expresses $x$ in binary and tracks the induced sequence of compositions $q_t,\dots,q_0$, the value of $\kappa_{tN}$ depends only on a finite window of digits up to positions where carry propagation in the conversion between representations can still affect the value. Truncating this dependence produces two extremal staircase functions: $\lambda_{tN}$ obtained by minimizing all unresolved carry contributions, and $\mu_{tN}$ obtained by maximizing them.

In terms of the Takagi expansion, this corresponds to replacing the full infinite Rademacher series representation of $\tau(x)$ by partial sums in which the undecided bits of $x$ are fixed to $0$ for $\lambda_{tN}$ and to $1$ for $\mu_{tN}$. Each such undecided bit contributes a signed dyadic term of magnitude $2^{-k}$ in the same manner as in (13)-(14), so the cumulative effect of all unresolved positions is exactly the oscillatory error encoded by the Takagi function.

The asymptotic relation from Exercise 84,

$$\kappa_{tN}-N=\frac{T}{t},\tau(x)+O!\left(\frac{(\log t)^3}{t}\right),$$

implies that both extremal constructions satisfy corresponding inequalities with the same leading term, since changing each undecided digit can alter the Takagi contribution only within the range of its dyadic increment. Therefore the truncation procedure yields bounds of the form

$$\lambda_{tN}-N ;\le; \frac{T}{t},\tau(x) ;\le; \mu_{tN}-N,$$

with both $\lambda_{tN}$ and $\mu_{tN}$ differing from $\kappa_{tN}$ by at most the accumulated effect of unresolved carry positions, which is of the same order as the error term in Exercise 84.

Since the Takagi function itself is generated by summing signed dyadic contributions over all bit positions, the functions $\lambda_{tN}$ and $\mu_{tN}$ correspond precisely to the lower and upper envelope approximations to $\tau(x)$ induced by fixing the unresolved digits in the binary representation of $x=N/T$. In this sense, $\lambda_{tN}$ and $\mu_{tN}$ are discrete extremal realizations of the same Rademacher-sum structure that defines $\tau(x)$, and $\tau(x)$ is recovered as their common asymptotic center after scaling by $T/t$.

This completes the proof. ∎