Small fractional parts of binary forms

2022/03/10 by Kiseok Yeon, Yeon, Kiseok
Mathematics · #Algebraic Geometry and Number Theory #Analytic Number Theory Research #FOS: Mathematics #Mathematical Approximation and Integration #Number Theory (math.NT)

paper · pdf · doi:10.48550/arxiv.2203.05535

openalex publication_date 2022/03/10 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We obtain bounds on fractional parts of binary forms of the shape Ψ(x,y)=α_k x^k+α_l x^ly^k-l+α_l-1x^l-1y^k-l+1+⋯+α₀ y^k with α_k,α_l,…,α₀∈ℝ and l≤ k-2. By exploiting recent progress on Vinogradov's mean value theorem and earlier work on exponential sums over smooth numbers, we derive estimates superior to those obtained hitherto for the best exponent σ, depending on k and l, such that min_\substack0≤ x,y≤ X
(x,y)≠ (0,0)‖Ψ(x,y)‖≤ X^-σ+ε.

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