论文标题
大型避免了无限序列的仿射副本
Large sets avoiding affine copies of infinite sequences
论文作者
论文摘要
ERDS的猜想指出，对于任何无限设置$ a \ subseteq \ Mathbb r $，存在$ e \ subseteq \ subseteq \ mathbb r $ for -lebesgue措施，不包含任何非平凡的封闭式副本。猜想对于大多数快速付费序列仍保持打开状态，包括几何序列$ a = \ {2^{ - k}：k \ geq 1 \} $。在本文中，我们考虑了$ {\ Mathbb r} $ in $ {\ Mathbb r} $中的无限减少序列$ a = \ {a_k：k \ geq 1 \} $，以规定的速率收敛到零；即$ \ log（a_n/a_ {n+1}）= e^{φ（n）} $，其中$φ（n）/n \ to 0 $ as as $ n \ to $ n \ to \ infty $。对数具有多项式衰减，尤其是几何序列的序列满足了这种情况。对于任何此类序列$ a $，我们构建了Hausdorff Dimension 1的Borel Set $ {\ Mathcal O} \ subseteq \ Mathbb r $ $，但是Lebesgue Meatuer Meate Zero却避免了$ A \ Cup \ cup \ {0 \ \} $的所有非平凡仿率。
A conjecture of Erdős states that for any infinite set $A \subseteq \mathbb R$, there exists $E \subseteq \mathbb R$ of positive Lebesgue measure that does not contain any nontrivial affine copy of $A$. The conjecture remains open for most fast-decaying sequences, including the geometric sequence $A = \{2^{-k} : k \geq 1\}$. In this article, we consider infinite decreasing sequences $A = \{a_k: k \geq 1\}$ in ${\mathbb R}$ that converge to zero at a prescribed rate; namely $\log (a_n/a_{n+1}) = e^{φ(n)} $, where $φ(n)/n\to 0$ as $n\to\infty$. This condition is satisfied by sequences whose logarithm has polynomial decay, and in particular by the geometric sequence. For any such sequence $A$, we construct a Borel set ${\mathcal O}\subseteq \mathbb R$ of Hausdorff dimension 1, but Lebesgue measure zero, that avoids all nontrivial affine copies of $A\cup\{0\}$.
