%0 Generic %1 sardari2015optimal %A Sardari, Naser Talebizadeh %D 2015 %K distributions quadratic-forms %T Optimal strong approximation for quadratic forms %U http://arxiv.org/abs/1510.00462 %X For a non-degenerate integral quadratic form $F(x_1, , x_d)$ in 5 (or more) variables, we prove an optimal strong approximation theorem. Fix any compact subspace $Ømega\subsetR^d$ of the affine quadric $F(x_1,\dots,x_d)=1$. Suppose that we are given a small ball $B$ of radius $0<r<1$ inside $Ømega$, and an integer $m$. Further assume that $N$ is a given integer which satisfies $N\gg(r^-1m)^4+\epsilon$ for any $\epsilon>0$. Finally assume that we are given an integral vector $(łambda_1, \dots, łambda_d) $ mod $m$. Then we show that there exists an integral solution $x=(x_1,\dots,x_d)$ of $F(x)=N$ such that $x_iłambda_i mod m$ and $xNB$, 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 $F(x_1, , x_4)$ in 4 variables we prove the same result if $N(r^-1m)^6+\epsilon$ and some non-singular local conditions for $N$ are satisfied. Based on some numerical experiments on the diameter of LPS Ramanujan graphs, we conjecture that the optimal strong approximation theorem holds for any quadratic form $F(X)$ in 4 variables with the optimal exponent $4$.

@misc{sardari2015optimal, abstract = {For a non-degenerate integral quadratic form $F(x_1, \dots , x_d)$ in 5 (or more) variables, we prove an optimal strong approximation theorem. Fix any compact subspace $\Omega\subset\mathbb{R}^d$ of the affine quadric $F(x_1,\dots,x_d)=1$. Suppose that we are given a small ball $B$ of radius $0<r<1$ inside $\Omega$, and an integer $m$. Further assume that $N$ is a given integer which satisfies $N\gg(r^{-1}m)^{4+\epsilon}$ for any $\epsilon>0$. Finally assume that we are given an integral vector $(\lambda_1, \dots, \lambda_d) $ mod $m$. 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 $F(x_1, \dots , x_4)$ in 4 variables we prove the same result if $N\geq (r^{-1}m)^{6+\epsilon}$ and some non-singular local conditions for $N$ are satisfied. Based on some numerical experiments on the diameter of LPS Ramanujan graphs, we conjecture that the optimal strong approximation theorem holds for any quadratic form $F(X)$ in 4 variables with the optimal exponent $4$.}, added-at = {2015-12-23T20:43:29.000+0100}, author = {Sardari, Naser Talebizadeh}, biburl = {https://www.bibsonomy.org/bibtex/225afecf6b19727dcf7ac2623ad4d373c/shabbychef}, description = {Optimal strong approximation for quadratic forms}, interhash = {c2a145e7574fbe77ff109cbc47cba394}, intrahash = {25afecf6b19727dcf7ac2623ad4d373c}, keywords = {distributions quadratic-forms}, note = {cite arxiv:1510.00462}, timestamp = {2015-12-23T20:43:29.000+0100}, title = {Optimal strong approximation for quadratic forms}, url = {http://arxiv.org/abs/1510.00462}, year = 2015 }