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Nonlinear uncertainty principle
Abstract
We prove an uncertainty principle for subsets of ℝ under weak branching assumptions. We impose a nonlinearity condition on the phase; without it our uncertainty principle fails e.g. for arithmetic progressions. The key ingredient in the proof is Shmerkin's nonlinear projection theorem. This paper is a part of a work in progress, but it is released earlier than the rest of that paper since it is also used in a paper of Cohen-Li-Tao. It also has an overlap with our recent paper [arXiv:2609.30261]; see the end of the introduction for more details.
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BibTeX
@misc{cohen2026nonlinear,
title = {{Nonlinear uncertainty principle}},
author = {Alex Cohen and Semyon Dyatlov},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.37549},
eprint = {2610.37549},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.37549},
}
RIS
TY - UNPB
TI - Nonlinear uncertainty principle
AU - Alex Cohen
AU - Semyon Dyatlov
PY - 2026
DA - 2026/10/27
PB - arXiv
DO - 10.48550/arxiv.2610.37549
UR - https://arxiv.org/abs/2610.37549
AB - We prove an uncertainty principle for subsets of ℝ under weak branching assumptions. We impose a nonlinearity condition on the phase; without it our uncertainty principle fails e.g. for arithmetic progressions. The key ingredient in the proof is Shmerkin's nonlinear projection theorem. This paper is a part of a work in progress, but it is released earlier than the rest of that paper since it is also used in a paper of Cohen-Li-Tao. It also has an overlap with our recent paper [arXiv:2609.30261]; see the end of the introduction for more details.
ER -