论文标题
蒙德里亚艺术问题的新结果
New results for the Mondrian art problem
论文作者
论文摘要
蒙德里安的问题包括将侧面长度的正方形$ n \ in \ nn $剖析到具有自然长度侧面的非统一矩形中，以使得$ d（n）$之间的差异$ d（n）$之间的矩形和最小的区域分区的平方是最小的。在本文中，我们根据方形分区的矩形数量计算$ d（n）$的一些界限。这些界限为我们提供了\ nn $中$ n \的某些值的最佳分区。我们提供一系列方形分区，以使$ d（n）/n^2 $趋于零$ n $大。 For the case of `perfect' partitions, that is, with $d(n)=0$, we show that, for any fixed powers $s_1,\ldots, s_m$, a square with side length $n=p_1^{s_1}\cdots p_m^{s_m}$, can have a perfect Mondrian partition only if $p_1$ satisfies a given lower bound.此外，如果$ n（x）$是$ x $（带有$ n \ le x $）的正方形的侧面长度，我们证明其“密度” $ \ frac {n（x）} {x）{x} $是$ \ frac {（frac {（frac frac {（frac log）{\ log（\ log log（x x）^2}）
The Mondrian problem consists of dissecting a square of side length $n\in \NN$ into non-congruent rectangles with natural length sides such that the difference $d(n)$ between the largest and the smallest areas of the rectangles partitioning the square is minimum. In this paper, we compute some bounds on $d(n)$ in terms of the number of rectangles of the square partition. These bounds provide us optimal partitions for some values of $n \in \NN$. We provide a sequence of square partitions such that $d(n)/n^2$ tends to zero for $n$ large enough. For the case of `perfect' partitions, that is, with $d(n)=0$, we show that, for any fixed powers $s_1,\ldots, s_m$, a square with side length $n=p_1^{s_1}\cdots p_m^{s_m}$, can have a perfect Mondrian partition only if $p_1$ satisfies a given lower bound. Moreover, if $n(x)$ is the number of side lengths $x$ (with $n\le x$) of squares not having a perfect partition, we prove that its `density' $\frac{n(x)}{x}$ is asymptotic to $\frac{(\log(\log(x))^2}{2\log x}$, which improves previous results.
