Improved convergence bounds for many slopes

Prove or otherwise establish the conjectured improvement of the bounds for \((R;s)\) when \(R=\{0,1,\infty,r_1,\ldots,r_k\}\) to the three-slope form \(2-c/D(R;s)\) for all \(k\), possibly after modifying the definition of rational complexity.

Background

Theorem 1.2 proves that, for a fixed number k+3k+3 of slopes, the sum-difference exponent approaches $2$ as the rational complexity D(R;s)D(R;s) grows, with a lower-bound correction of order log⁡(2+D)/D\log(2+D)/D and an upper-bound correction of order D−(k+1)D^{-(k+1)}.

The author explicitly conjectures that the weaker many-slope rate can be sharpened to the same order $1/D$ established in the three-slope case, potentially with a slight adjustment to rational complexity.

References

We tentatively conjecture that the bounds in sam can be improved to be of the form 2ab for all $k$, not just $k=0$ (possibly after some slight adjustments to the definition of rational complexity).

sam:

2−Cklog⁡(2+D(R;s))D(R;s)≤(R;s)≤2−ckD(R;s)k+12 - \frac{C_k \log(2+D(R;s))}{D(R;s)} \leq (R; s) \leq 2 - \frac{c_k}{D(R;s)^{k+1}}

2ab:

2−c2D(R;s)≤(R;s)≤2−c1D(R;s)2 - \frac{c_2}{D(R;s)} \leq \left(R; s\right) \leq 2 - \frac{c_1}{D(R;s)}

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