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A partial answer. It is clear that we should consider only $\alpha,\beta>1$. Suppose we want to find $n$ such that $n\in T_\beta$, $n\notin T_\alpha$. It is sufficient to find integers $l$ and $k$ s …

If $\frac{p_{2k}}{ q_{2k}}$ and $\frac{p_{2k+1}}{ q_{2k+1}}$ ($k\ge 0$) are consecutive convergents of the continued fraction expansion of $x$ then $$\left|x-\frac{p_{2k}}{ q_{2k}}\right|+\left|x-\fra …

You can find this expansion in the book Lorentzen L. & Waadeland H. Continued fractions with applications North-Holland Publishing Co., 1992 (formula (4.3.10)). As I understand, this document is a mor …

You can find the answer in the paper Mark C. Wilson, Diagonal asymptotics for products of combinatorial classes: $$\sum_{k=0}^n\binom{n}{k}^d\sim\sqrt{\frac{2^{d-1}}{d}}\frac{2^{dn}}{(\pi n)^{\frac{d- …

Stationary phase method (in the usual setup) gives asymptotic for $$ I(\lambda)=\int_{a}^{b} f(t) e^{i \lambda \varphi(t)} d t, $$ when at any stationary point $x_0$ ($\varphi'(x_0)=0$) second deriva …