Optimal strong approximation for quadratic forms
Optimal strong approximation for quadratic forms
复制标题
二次形式的最优强近似
DOI：
10.1215/00127094-2019-0007
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发表时间：
2015
影响因子：
2.5
通讯作者：
Naser T. Sardari
中科院分区：
文献类型：
--
作者：
Naser T. Sardari
对于非退化积分二次型$F(x_1，\dots，x_d)$，证明了一个最佳强逼近定理。设$Omega$是实数上仿射二次$F(x_1，\dots，x_d)=1$的固定紧子集.取一个半径为$0$的小球$B$。最后假定给出了一个积分向量$(\lambda_1，\dots，\lambda_d)$mod$m$。然后证明了在满足所有局部条件的条件下，存在$F(X)=N$的积分解$X=(x_1，\dots，x_d)$，使得$x_i\EQUEV\lambda_i\Text{mod}m$和$\frac{X}{\Sqrt{N}}\在B$中。我们还证明了4是最好的可能指数。此外，对于四元非退化积分二次型，证明了当$N$为奇数且$N为奇数，且$N，Omega}(r^-1}m)^{6+\epsilon}$时，得到了同样的结果。基于我们对LPSRamanujan图的直径和注解中出现的特定和的期望平方根抵消的数值实验，我们猜想该定理对任何具有最优指数$4$的四变量二次型都成立。
For a non-degenerate integral quadratic form $F(x_1, \dots , x_d)$ in $d\geq5$ variables, we prove an optimal strong approximation theorem. Let $\Omega$ be a fixed compact subset of the affine quadric $F(x_1,\dots,x_d)=1$ over the real numbers. Take a small ball $B$ of radius $0 0$. Finally assume that an integral vector $(\lambda_1, \dots, \lambda_d) $ mod $m$ is given. Then we show that there exists an integral solution $X=(x_1,\dots,x_d)$ of $F(X)=N$ such that $x_i\equiv \lambda_i \text{ mod } m$ and $\frac{X}{\sqrt{N}}\in B$, provided that all the local conditions are satisfied. We also show that 4 is the best possible exponent. Moreover, for a non-degenerate integral quadratic form in 4 variables we prove the same result if $N$ is odd and $N\gg_{\delta,\Omega} (r^{-1}m)^{6+\epsilon}$. Based on our numerical experiments on the diameter of LPS Ramanujan graphs and the expected square root cancellation in a particular sum that appears in Remark~\ref{evidence}, we conjecture that the theorem holds for any quadratic form in 4 variables with the optimal exponent $4$.
影响因子：
1
作者：
Ghosh A
通讯作者：
Ghosh A