论文标题
没有流行差异的模式
Patterns without a popular difference
论文作者
论文摘要
哪个有限设置$ p \ subseteq \ mathbb {z}^r $ with $ | p | \ ge 3 $具有以下属性：对于每个$ a \ subseteq [n]^r $，都有一些非零整数$ d $，因此$ a $ conting $（α^{| p |} - o（o（1））n^r $翻译的$ d \ cdot p = \ cdot p = \ c p = \ p p \ p p \ p \ n $ n $ n $ n $， 格林表明所有3点$ p \ subseteq \ mathbb {z} $具有以上属性。 Green和Tao表明，表格$ p = \ {a+b，a+c，a+b+c \} \ subseteq \ mathbb {z} $的四点组也具有属性。我们表明，没有其他集合具有上述属性。此外，对于各种$ p $，我们在可以保证找到的$ d \ cdot p $的翻译数量上提供了新的上限。
Which finite sets $P \subseteq \mathbb{Z}^r$ with $|P| \ge 3$ have the following property: for every $A \subseteq [N]^r$, there is some nonzero integer $d$ such that $A$ contains $(α^{|P|} - o(1))N^r$ translates of $d \cdot P = \{d p : p \in P\}$, where $α= |A|/N^r$? Green showed that all 3-point $P \subseteq \mathbb{Z}$ have the above property. Green and Tao showed that 4-point sets of the form $P = \{a, a+b, a+c, a+b+c\} \subseteq \mathbb{Z}$ also have the property. We show that no other sets have the above property. Furthermore, for various $P$, we provide new upper bounds on the number of translates of $d \cdot P$ that one can guarantee to find.
