Asymptotic equality in the continuous-limit formula

Prove that, for coprime integers \(a,b\) with \(b\to\infty\) and \(a/b\to\alpha\), the upper bound \(\left(\{0,1,\infty\};a/b\right)\ge 2-(c_\alpha+o(1))/\log b\) is an asymptotic equality, so that the leading asymptotic behavior is governed by the variational quantity \(c_\alpha\).

Background

The continuous-limit theorem gives a lower bound for the sum-difference exponent associated with the slope a/ba/b, with a coefficient cαc_\alpha defined through an infimum of a differential-entropy functional over smooth compactly supported probability densities.

The paper establishes only the lower bound and then explicitly conjectures that it is sharp to first order in 1/log⁡b1/\log b. If true, this would show that rational complexity determines the logarithmic scale while the variational problem determines the leading coefficient.

References

We tentatively conjecture that the upper bound in up is in fact an asymptotic equality, so that the asymptotic behavior of $ \left({0,1,\infty}, \frac{a}{b}\right)$ is controlled not only by the rational complexity (as represented by the $\log b$ denominator), but also by the variational quantity $c_\alpha$ appearing in the numerator.

up:

({0,1,∞};ab)≥2−cα+o(1)log⁡b\left(\{0,1,\infty\};\frac{a}{b}\right) \geq 2 - \frac{c_\alpha+o(1)}{\log b}

— Sum-difference exponents for boundedly many slopes, and rational complexity  (2511.15135 - Tao, 19 Nov 2025) in Section 1.2, immediately after Theorem 1.3 (“Continuous limit”)