The Coin Exchange Problem and the Structure of Cube Tilings

TL;DR

This paper explores the relationship between a combinatorial problem called the Coin Exchange Problem and the structure of periodic cube tilings, proposing a conjecture about the representation of certain sets.

Contribution

It introduces a conjecture linking the size of specific subsets to linear combinations of integers and connects this to the structure of periodic cube tilings.

Findings

Conjecture relating the size of set D to linear combinations of k_i.

Connection established between the conjecture and the structure of cube tilings.

Insights into the combinatorial and geometric aspects of the problem.

Abstract

Let k_1,...,k_d be positive integers, and D be a subset of [k_1]x...x[k_d], whose complement can be decomposed into disjoint sets of the form {x_1}x...x{x_{s-1}}x[k_s]x{x_{s+1}}x...x{x_d}. We conjecture that the number of elements of D can be represented as a linear combination of the numbers k_1,..., k_d with non-negative integer coefficients. A connexion of this conjecture with the structure of periodical cube tilings is revealed.