1 results
Perfectly packing a cube by cubes of nearly harmonic sidelength
- Part of
-
- Journal:
- Canadian Mathematical Bulletin / Volume 66 / Issue 3 / September 2023
- Published online by Cambridge University Press:
- 14 February 2023, pp. 1061-1071
- Print publication:
- September 2023
-
- Article
-
- You have access
- HTML
- Export citation
-
Let d be an integer greater than
$1$, and let t be fixed such that 
$\frac {1}{d} < t < \frac {1}{d-1}$. We prove that for any 
$n_0$ chosen sufficiently large depending on t, the d-dimensional cubes of sidelength 
$n^{-t}$ for 
$n \geq n_0$ can perfectly pack a cube of volume 
$\sum _{n=n_0}^{\infty } \frac {1}{n^{dt}}$. Our work improves upon a previously known result in the three-dimensional case for when 
$\frac {1}{3} < t \leq \frac {4}{11} $ and 
$n_0 = 1$ and builds upon recent work of Terence Tao in the two-dimensional case.
