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Almost k-th powers in short intervals
Abstract
Let k≥ 3 be a fixed integer and x be a large real number. Let 0<c0<θ≤ 1 be a fixed real number for some c0>0. In this paper we show that there exists a constant 1/2≤ δk(θ)<1 such that the interval [x, x + xδk(θ) +ε ] contains an integer of the form n1n2 ⋯ nk such that |nj−n1/k|≪ nθ/k (j=1,2, ⋯, k). Especially we have δ3(1) = δ4(1) = 1/2, which improves previous results of Chan.
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BibTeX
@misc{zhai2026almost,
title = {{Almost $k$-th powers in short intervals}},
author = {Wenguang Zhai},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.00386},
eprint = {2609.00386},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.00386},
}
RIS
TY - UNPB
TI - Almost k-th powers in short intervals
AU - Wenguang Zhai
PY - 2026
DA - 2026/09/01
PB - arXiv
DO - 10.48550/arxiv.2609.00386
UR - https://arxiv.org/abs/2609.00386
AB - Let k≥ 3 be a fixed integer and x be a large real number. Let 0<c_0<θ≤ 1 be a fixed real number for some c_0>0. In this paper we show that there exists a constant 1/2≤ δ_k(θ)<1 such that the interval [x, x + x^δ_k(θ) +ε ] contains an integer of the form n_1n_2 ⋯ n_k such that |n_j−n^1/k|≪ n^θ/k (j=1,2, ⋯, k). Especially we have δ_3(1) = δ_4(1) = 1/2, which improves previous results of Chan.
ER -