Cubes in the Torus
Abstract
For $q> p$, let $T(n,q,p)$ be the minimum number of translates of the cube \(\{0,1,\dots,p-1\}^n\) required to cover the $n$-dimensional torus $(\mathbb{Z}/q\mathbb{Z})^n$. We show that for each $q$ there exists a constant $1\le Λ_q \le 2$ such that $T(n,q,2)=(Λ_q + o(1))(q/2)^n$.
Disclosure
“2 Statement on the use of AI Generative AI tools were used for all the proofs in this paper, which also provided a Lean formalisation [7]. All arguments were subsequently checked, edited, and presented by the authors, who take full responsibility for the correctness of the results.”
PDF page 2
- Classification
- Substantial proof generation
- Multiplier
- 10
- Verified
Structural counts
Pages 6 pdf
Theorems 3 source
Lemmas 3 source
Propositions 1 source
Corollaries 0 source
Definitions 0 source
Displayed equations 26 source
Bibliography entries 8 source
Appendix pages 0 estimated
Count notes
- Source counts use the expanded primary TeX file main.tex.
- Appendix pages include the first PDF page with an explicit Appendix heading through the final page.