PreprintCC BY 4.0arXiv
An Approximate Version of Vu's Theorem on Economical Subbases For Non-Integer Exponents
Abstract
We prove an approximate analog of Vu's Theorem on economical bases of kth powers for non-integer powers. Fix a non-integer θ> 2 and real τ> 0. Let $s \geq θ^2+9θ^{3/2}+2 $ if θ> 3 and s ≥ (⌊ 2θ⌋+1)(⌊ 2θ⌋+2)+1 if 2 < θ< 3. We show the existence of a set X ⊆ ℕ such that the number RX,s,θ,τ(Λ) of integer solutions (x1,…,xs) ∈ Xs to the equation |x1θ +⋯+ xsθ − Λ| < τ satisfies RX,s,θ,τ(Λ) asymp τlog(Λ) for all sufficiently large real Λ> 0.
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BibTeX
@misc{bhargava2026approximate,
title = {{An Approximate Version of Vu's Theorem on Economical Subbases For Non-Integer Exponents}},
author = {Ataleshvara Bhargava},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.27823},
eprint = {2609.27823},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.27823},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - An Approximate Version of Vu's Theorem on Economical Subbases For Non-Integer Exponents
AU - Ataleshvara Bhargava
PY - 2026
DA - 2026/09/01
PB - arXiv
DO - 10.48550/arxiv.2609.27823
UR - https://arxiv.org/abs/2609.27823
AB - We prove an approximate analog of Vu's Theorem on economical bases of kth powers for non-integer powers. Fix a non-integer θ> 2 and real τ> 0. Let $s \geq θ^2+9θ^{3/2}+2 $ if θ> 3 and s ≥ (⌊ 2θ⌋+1)(⌊ 2θ⌋+2)+1 if 2 < θ< 3. We show the existence of a set X ⊆ ℕ such that the number R_X,s,θ,τ(Λ) of integer solutions (x_1,…,x_s) ∈ X^s to the equation |x_1^θ +⋯+ x_s^θ − Λ| < τ satisfies R_X,s,θ,τ(Λ) asymp τlog(Λ) for all sufficiently large real Λ> 0.
ER -