# Weighted $1\times1$ cut-and-project sets in bounded distance to a lattice

## TL;DR

This paper demonstrates that certain weighted cut-and-project sets in one dimension are bounded distance equivalent to a lattice under specific smoothness conditions on the weight function.

## Contribution

It extends previous results by establishing bounded distance equivalence for weighted cut-and-project sets with particular regularity conditions on the weight function.

## Key findings

- Weighted cut-and-project sets are bounded distance equivalent to a lattice under certain conditions.
- Continuity, piecewise linearity, or bounded curvature of the weight function ensure bounded distance equivalence.
- Results apply to one-dimensional physical and internal spaces.

## Abstract

Recent results of Grepstad and Lev are used to show that weighted cut-and-project sets with one-dimensional physical space and one-dimensional internal space are bounded distance equivalent to some lattice if the weight function $h$ is continuous on the internal space, and if $h$ is either piecewise linear, or twice differentiable with bounded curvature.