Necessary conditions for binomial collisions

We shall give some necessary conditions for the equation $\binom{x}{a}=\binom{y}{b}$ to hold: if $\binom{2n+δ}{n-m}=\binom{2n+l}{n-k}$ with $δ=0$ or $1$, $0<m\leq 0.735k$, $k<n$ and $n$ sufficiently large, then $l>(cn/\log n)^{40/21}$ for some constant $c$.

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