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The additive monoid $R_+(x)$ is defined as the set of all nonnegative integerlinear combinations of binomial coefficients $\binom{x}{n}$ for $n \in \mathbbZ_+$. This paper is concerned with the inquiry into the structure of$R_+(\alpha)$ for complex numbers $\alpha.$ Particularly interesting is thecase of algebraic $\alpha$ which are not non-negative integers. This questionis motivated by the study of functors between Deligne categories$\textrm{Rep}(S_t)$ (and also $\textrm{Rep}(\textrm{GL}_t)$) for $t \in \mathbbC\backslash \mathbb Z_+$. We prove that this object is a ring if and only if$\alpha$ is an algebraic number that is not a nonnegative integer. Furthermore,we show that all algebraic integers generated by $\alpha,$ i.e. all elements of$\mathcal O_{\mathbb Q(\alpha)},$ are also contained in this ring. We also givetwo explicit representations of $R_+(\alpha)$ for both algebraic integers andgeneral algebraic numbers $\alpha.$ One is in terms of inequalities for thevaluations with respect to certain prime ideals and the other is in terms ofexplicitly constructed generators. We show how these results work in thecontext of the study of symmetric monoidal functors between Deligne categoriesin positive characteristic. Moreover, this leads to a particularly simpledescription of $R_+(\alpha)$ for both quadratic algebraic numbers and roots ofunity.

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