Sums of binomial coefficients evaluated at $\alpha \in \overline{\mathbb{Q}},$ and applications

Daniil Kalinov, Andrei Mandelshtam

The additive monoid $R_{+}(x)$ is defined as the set of all nonnegative integer linear combinations of binomial coefficients $\left(\genfrac{}{}{0ex}{}{x}{n}\right)$ for $n\in {\mathbb{Z}}_{+}$. This paper is con…