Bounded step functions and factorial ratio sequences

TL;DR

This paper investigates step functions linked to factorial ratio sequences, providing bounds and restrictions that help determine when these sequences are integral, with applications to classifying cyclic quotient singularities.

Contribution

It introduces a lower bound for the mean square of certain step functions, offering new criteria for the integrality of factorial ratio sequences.

Findings

Established a lower bound for the mean square of step functions.

Derived restrictions on the integrality of factorial ratio sequences.

Connected the results to the classification of cyclic quotient singularities.

Abstract

We study certain step functions whose nonnegativity is related to the integrality of sequences of ratios of factorial products. In particular, we obtain a lower bound for the mean square of such step functions which allows us to give a restriction on when such a factorial ratio sequence can be integral. Additionally, we note that this work has applications to the classification of cyclic quotient singularities.