著名的 Erdös-Szemerédi 加乘法估計定理說明了任意的有限整數子集合的加法集或是乘法集的個數會有擴張現象。在這篇論文裡，我們探討一些此類估計的變形。我們證明了給定實數裡的任意一個有限子集，它的加法集或是平方加法集也會有擴張現象。我們證明的工具包含了離散幾何的一些定理以及加法組合學的一些技巧。;The well-known sum-product estimate of Erdös and Szemerédi asserts that any finite set in integers either has sum set or product set much larger than itself. In this thesis, we also study a variant of sum-product estimate in reals. We prove that for any finite set in reals either its sum set or sum of squares set is large. Our tools include some theorems from incidences geometry and additive combinatorics.
