Abstract
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 $4$.
Description
Optimal strong approximation for quadratic forms
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