论文标题
在不同几何形式的有限套件中，最大数量的$ 3 $ - 学期算术进展
The maximal number of $3$-term arithmetic progressions in finite sets in different geometries
论文作者
论文摘要
Green和Sisask表明，$ N $ element Integers中的$ 3 $ term算术进程的最大数量为$ \ lceil n^2/2 \ rceil $;很容易看出，如果整数由真实的线路或任何欧几里得空间代替，则相同。我们在一般度量空间中研究了这个问题，其中三重$（a，b，c）$在公制空间中的点被认为是$ 3 $ - term算术算术的进程，如果$ d（a，b）= d（a，b）= d（b，c）= \ frac {1} {1} {1} {2} {2} {2} d（a，c）$。特别是，我们表明，绿色和西萨斯的结果延伸到任何cartan cartan歧管（尤其是双曲线空间），但对于任何$ r \ geq 3 $，但不存在于球形几何形状或$ r $ toem-regular-regular-regular-regular-regular tree。
Green and Sisask showed that the maximal number of $3$-term arithmetic progressions in $n$-element sets of integers is $\lceil n^2/2\rceil$; it is easy to see that the same holds if the set of integers is replaced by the real line or by any Euclidean space. We study this problem in general metric spaces, where a triple $(a,b,c)$ of points in a metric space is considered a $3$-term arithmetic progression if $d(a,b)=d(b,c)=\frac{1}{2}d(a,c)$. In particular, we show that the result of Green and Sisask extends to any Cartan--Hadamard manifold (in particular, to the hyperbolic spaces), but does not hold in spherical geometry or in the $r$-regular tree, for any $r\geq 3$.
