论文标题
Hausdorff尺寸的ABC总产量问题范围
Hausdorff dimension bounds for the ABC sum-product problem
论文作者
论文摘要
本文的目的是完成以下结果的证明。令$ 0 <β\ leqα<1 $和$κ> 0 $。然后，存在$η> 0 $，以便每当$ a，b \ subset \ mathbb {r} $都是borel集，带有$ \ dim _ {\ mathrm {\ mathrm {h}} a =α$ and $ \ \ \ dim _ { \ {c \ in \ Mathbb {r}：\ dim _ {\ mathrm {h}}（a + cb）\ leqleqα +η\} \ leq \ leq \ tfrac {α-β} {α-β} {1-β} +κ。 本文是作者以前的2021年作品的续集，该续集大概是用$ \ dim _ {\ mathrm {h}}}（a + cb）$建立了相同的结果，被$ \ dim _ {\ dim _ {\ mathrm {b}}}（a + cb）（a + cb）$替换而成。事实证明，在$δ$ dist的语句的水平上，表面上较弱的盒子尺寸结果正式意味着Hausdorff尺寸结果。
The purpose of this paper is to complete the proof of the following result. Let $0 < β\leq α< 1$ and $κ> 0$. Then, there exists $η> 0$ such that whenever $A,B \subset \mathbb{R}$ are Borel sets with $\dim_{\mathrm{H}} A = α$ and $\dim_{\mathrm{H}} B = β$, then $$\dim_{\mathrm{H}} \{c \in \mathbb{R} : \dim_{\mathrm{H}} (A + cB) \leq α+ η\} \leq \tfrac{α- β}{1 - β} + κ.$$ This extends a result of Bourgain from 2010, which contained the case $α= β$. This paper is a sequel to the author's previous work from 2021 which, roughly speaking, established the same result with $\dim_{\mathrm{H}} (A + cB)$ replaced by $\dim_{\mathrm{B}}(A + cB)$, the box dimension of $A + cB$. It turns out that, at the level of $δ$-discretised statements, the superficially weaker box dimension result formally implies the Hausdorff dimension result.
