On residues of rounded shifted fractions with a common numerator

TL;DR

This paper studies the distribution of residues in shifted fractional sequences, deriving formulas for their counts in specific congruence classes and connecting these to lattice point problems in algebraic number rings.

Contribution

It introduces exact and asymptotic formulas for residue counts in shifted fractional sequences and links these to lattice point problems in algebraic number theory.

Findings

Formulas for the number of sequence elements in given residue classes

Connections between residue counts and lattice points in conic regions

Applications to Gaussian and Eisenstein integers

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.