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A Borsuk-Ulam theorem for adaptive continuous measurements
Abstract
We prove a Borsuk-Ulam theorem for adaptive measurements: if 2n≤ m, then any n adaptively chosen continuous functionals take identical values at two antipodal points of any sphere in ℝm. Consequently, at least ⌈log2(m+1)⌉ adaptive continuous measurements are needed to learn the vectors of a ball in ℝm with an error smaller than its radius, which matches the recently obtained upper bound up to one measurement. The bound extends to Bernstein widths in Banach spaces. The main idea of the proof was found by Claude Fable 5, an AI model developed by Anthropic, and we document the development in a last section.
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BibTeX
@misc{ullrich2026borsukulam,
title = {{A Borsuk-Ulam theorem for adaptive continuous measurements}},
author = {Mario Ullrich},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.33781},
eprint = {2610.33781},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.33781},
}
RIS
TY - UNPB
TI - A Borsuk-Ulam theorem for adaptive continuous measurements
AU - Mario Ullrich
PY - 2026
DA - 2026/10/24
PB - arXiv
DO - 10.48550/arxiv.2610.33781
UR - https://arxiv.org/abs/2610.33781
AB - We prove a Borsuk-Ulam theorem for adaptive measurements: if 2^n≤ m, then any n adaptively chosen continuous functionals take identical values at two antipodal points of any sphere in ℝ^m. Consequently, at least ⌈log_2(m+1)⌉ adaptive continuous measurements are needed to learn the vectors of a ball in ℝ^m with an error smaller than its radius, which matches the recently obtained upper bound up to one measurement. The bound extends to Bernstein widths in Banach spaces. The main idea of the proof was found by Claude Fable 5, an AI model developed by Anthropic, and we document the development in a last section.
ER -