Define $S(k,2n)=\sum_{i=-k}^k\binom{2n}{n+i}$ at every $k\in\{0,\dots,n\}$.

We know $$\ln(S(\gamma\mbox{ } n^\gamma,2n))\asymp(2n\ln2-\frac12\ln\pi-\frac12\ln n)$$ at $\gamma\rightarrow0$ and $$\ln(S(\gamma\mbox{ } n^\gamma,2n))=2n\ln 2$$ at $\gamma\rightarrow1$.

Is there a good approximating expression that interpolates for $$\ln(S(\gamma\mbox{ } n^\gamma,2n))$$ at general $\gamma\in(0,1)$ between $\gamma=0$ and $\gamma=1$ here in the sense that gives information on at what $\gamma$ we may expect $2n\ln2 - f(n)$ for a given $f(n)<\frac12\ln\pi+\frac12\ln n$?

Denote $\gamma n^{\gamma}=m^{\beta}$ and $2n=m^{\alpha}$. What is the right description when $\beta<\frac\alpha2$ and $\frac\alpha2 - \frac1{c\log m}<\beta$ holds at a fixed $c>0$? Does $$m^\alpha\ln2 - (\frac\alpha2-\beta)\ln m -\frac12\ln\frac\pi2$$ seem an accurate estimate?

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