论文标题
通过几乎谐波侧长的正方形完美地包装正方形
Perfectly packing a square by squares of nearly harmonic sidelength
论文作者
论文摘要
Meir和Moser的一个众所周知的开放问题询问Sidelength $ 1/N $的平方是否可以完美地包装到面积$ \ sum_ {n = 2}^\ infty \ frac {1} {1} {n^2} {n^2} = \ frac {frac {frac {frac {frac {frac {frac {frac {frac {frac {frac {frac {frac {frac {frac {frac {frac）=在本文中，我们表明，对于任何$ 1/2 <t <1 $，以及任何$ n_0 $，取决于$ t $，sideLength $ n^{ - t} $的squares for $ n \ geq n_0 $都可以完美地包装到一个面积$ \ sum_ $ \ sum_ \ sum_ {n_0}^n_0}^n_0}^n_0}^\ uffty fty for \ frac {1} {n^{2t}} $。以前以$ 1/2 <t \ leq 2/3 $（在这种情况下，可以服用$ n_0 = 1 $），以前以$ 1/2 <t \ leq 2/3 $包装矩形而不是正方形。
A well known open problem of Meir and Moser asks if the squares of sidelength $1/n$ for $n \geq 2$ can be packed perfectly into a square of area $\sum_{n=2}^\infty \frac{1}{n^2} = \frac{π^2}{6}-1$. In this paper we show that for any $1/2 < t < 1$, and any $n_0$ that is sufficiently large depending on $t$, the squares of sidelength $n^{-t}$ for $n \geq n_0$ can be packed perfectly into a square of area $\sum_{n=n_0}^\infty \frac{1}{n^{2t}}$. This was previously known (if one packs a rectangle instead of a square) for $1/2 < t \leq 2/3$ (in which case one can take $n_0=1$).
