Effective density of values of indefinite ternary inhomogeneous quadratic forms

2024/08/06 by Kelmer, Dubi · 1 citation
#FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.2408.03153

Abstract

Given an inhomogeneous quadratic form Q_ξ(v)=Q(v+ξ) with Q an indefinite ℚ-isotropic rational ternary form and ξ∈ ℝ³ irrational, we prove an effective lower bound for the number of integer vectors v∈ ℤⁿ with ‖v‖ ≤ T such that |Q_ξ(v)-t|<δ that is valid for any t∈ ℝ and all δ≥ T^-ν, with ν>0 depending explicitly on the Diophantine properties of ξ. In particular, for ξ with algebraic entries we can take any ν<(1)/(8).

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• A survey and a result on inhomogeneous quadratic forms

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