论文标题
凸性，挤压和Elekes-Szabó定理
Convexity, Squeezing, and the Elekes-Szabó Theorem
论文作者
论文摘要
本文探讨了凸度和总和之间的关系。特别是，我们表明，基本数字理论方法（主要是挤压原理的应用）可以使用Elekes-Szabó定理来增强，以提供新的信息。也就是说，如果我们让$ a \ subset \ mathbb r $，我们证明存在$ a，a'\ in a $ so \ [\ left | \ frac {（aa+1）^{（2）}（a'a+1）^{（2）}}} {（aa+1）^{（2）}（a'a+1）} \ right | \gtrsim |A|^{31/12}.\] We are also able to prove that \[ \max \{|A+A-A|, |A^2+A^2-A^2|, |A^3 + A^3 - A^3|\} \gtrsim |A|^{19/12}.\] Both of these bounds are improvements of recent results and takes advantage of computer代数处理一些计算。
This paper explores the relationship between convexity and sum sets. In particular, we show that elementary number theoretical methods, principally the application of a squeezing principle, can be augmented with the Elekes-Szabó Theorem in order to give new information. Namely, if we let $A \subset \mathbb R$, we prove that there exist $a,a' \in A$ such that \[\left | \frac{(aA+1)^{(2)}(a'A+1)^{(2)}}{(aA+1)^{(2)}(a'A+1)} \right | \gtrsim |A|^{31/12}.\] We are also able to prove that \[ \max \{|A+A-A|, |A^2+A^2-A^2|, |A^3 + A^3 - A^3|\} \gtrsim |A|^{19/12}.\] Both of these bounds are improvements of recent results and takes advantage of computer algebra to tackle some of the computations.
