论文标题
在利特伍德的猜想中的整数数量的下限
On a lower bound of the number of integers in Littlewood's conjecture
论文作者
论文摘要
我们表明，对于任何$ 0 <γ<1/2 $，除了hausdorff dimension in \sqrtγ$，任何$ \sqrtγ$，任何$ 0 $ 0 <\ varepsilon <1 $和任何大的$ n \ in \ n iN \ mathbb n} $ n} $ n} $ n数字，那个$ n \ langlenα\ rangle \ langlenβ\ rangle <\ varepsilon $大于$γ\ varepsilon \ log n $ n $ to均匀常数。这可以看作是一个定量结果，这是一个事实，即特殊的猜想的特殊设置具有Hausdorff Dimension Zero，这是M. Einsiedler，A。Katok和E. Lindenstrauss在2000年代获得的。为了证明证明，我们研究了对$ \ rm {sl}（3，\ mathbb {r}）/\ rm {sl}（3，\ mathbb {z}）$的对角行动的行为，并表明我们可以对Littlewood的$（ybe）进行定量的结果，并表明我们可以对$（ybe）进行定量的量表。我们还估计，特殊集合的Hausdorff尺寸很小。
We show that, for any $0<γ<1/2$, any $(α,β)\in\mathbb{R}^2$ except on a set with Hausdorff dimension about $\sqrtγ$, any small $0<\varepsilon<1$ and any large $N\in\mathbb{N}$, the number of integers $n\in[1,N]$ such that $n\langle nα\rangle\langle nβ\rangle<\varepsilon$ is greater than $γ\varepsilon\log N$ up to a uniform constant. This can be seen as a quantitative result on the fact that the exceptional set to Littlewood's conjecture has Hausdorff dimension zero, obtained by M. Einsiedler, A. Katok and E. Lindenstrauss in 2000's. For the proof, we study the behavior of the empirical measures with respect to the diagonal action on $\rm{SL}(3,\mathbb{R})/\rm{SL}(3,\mathbb{Z})$ and show that we can obtain a quantitative result on Littlewood's conjecture for $(α,β)$ if the corresponding empirical measures are well-behaved. We also estimate Hausdorff dimension of the exceptional set to be small.
