On residues of rounded shifted fractions with a common numerator

Abstract

For any positive integer $n$ along with parameters $\alpha$ and $\nu$, we define and investigate $\alpha$-shifted, $\nu$-offset, floor sequences of length $n$. We find exact and asymptotic formulas for the number of integers in such a sequence that are in a particular congruence class. As we will see, these quantities are related to certain problems of counting lattice points contained in regions of the plane bounded by conic sections. We give specific examples for the number of lattice points contained in elliptical regions and make connections to a few well-known rings of integers, including the Gaussian integers and Eisenstein integers.