Distribution of sums of square roots modulo 1

2024/04/01 by Iyer, Siddharth
#11J71 #FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.2404.01069

Abstract

We improve upon a result of Steinerberger (2024) by demonstrating that for any fixed k ∈ ℕ and sufficiently large n, there exist integers 1 ≤ a₁, …, a_k ≤ n satisfying: 0 lt; ‖ ∑_j=1^k √(a_j) ‖ = O(n^-k/2). The exponent k/2 improves upon the previous exponent of c k^1/3 of Steinerberger (2024), where c>0 is an absolute constant. We also show that for α∈ ℝ, there exist integers 1 ≤ b₁, …, b_k ≤ n such that: ‖ ∑_j=1^k √(b_j) - α‖ = O(n^-γ_k), where γ_k ≥ (k-1)/(4) and γ_k = k/2 when k=2^m - 1, m=1,2,…. Importantly, our approach avoids the use of exponential sums.

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