## Abstract

We estimate double sums S_{χ}(a, I, G) = ∑_{x∈I}∑_{λ∈G}χ(x + aλ), 1 ≤ a < p - 1, with a multiplicative character χ modulo p where I = {1, . . . , H} and G is a subgroup of order T of the multiplicative group of the finite field of p elements. A nontrivial upper bound on S _{χ} (a, I, G) can be derived from the Burgess bound if H ≥ p^{1/4+ε} and from some standard elementary arguments if T ≥ p^{1/2+ε}, where ε > 0 is arbitrary. We obtain a nontrivial estimate in a wider range of parameters H and T . We also estimate double sums T_{χ}(a, G) = ∑_{χ,μ∈G} χ(a + λ + μ), 1 ≤ a < p - 1, and give an application to primitive roots modulo p with three nonzero binary digits.

Original language | English |
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Pages (from-to) | 376-390 |

Number of pages | 15 |

Journal | Bulletin of the Australian Mathematical Society |

Volume | 90 |

Issue number | 3 |

DOIs | |

Publication status | Published - 2 Dec 2014 |

Externally published | Yes |

## Keywords

- Character sums
- Intervals
- Multiplicative subgroups of finite fields