## Abstract

For a nondegenerate integral quadratic form F.x_{1};:::;x_{d}/ in d 5 variables, we prove an optimal strong approximation theorem. Let be a fixed compact subset of the affine quadric F.x_{1};:::;x_{d}/ D 1 over the real numbers. Take a small ball B of radius 0 < r < 1 inside , and an integer m. Further assume that N is a given integer which satisfies N _{ı;} .r^{1}m/^{4}C^{ı} for any ı > 0. Finally assume that an integral vector ._{1};:::;_{d}/ mod m is given. Then we show that there exists an integral solution x D .x_{1};:::;x_{d}/ of F.x/ D N such that x_{i i} mod m and ^{px} _{N} 2 B, provided that all the local conditions are satisfied. We also show that 4 is the best possible exponent. Moreover, for a nondegenerate integral quadratic form in four variables, we prove the same result if N is odd and N _{ı;} .r^{1}m/^{6}C. 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 6.8, we conjecture that the theorem holds for any quadratic form in four variables with the optimal exponent 4.

Original language | English (US) |
---|---|

Pages (from-to) | 1887-1927 |

Number of pages | 41 |

Journal | Duke Mathematical Journal |

Volume | 168 |

Issue number | 10 |

DOIs | |

State | Published - 2019 |

## All Science Journal Classification (ASJC) codes

- Mathematics(all)