PreprintCC BY 4.0arXiv
Sumsets of Ahlfors--David regular sets
Abstract
We prove that there is an absolute constant c>0 such that for every finite (α,M)-Ahlfors--David regular set A⊆[N] with 0≤α<1 and M≥2, we have the sumset estimate |A+A|≥ |A|1+c(1−α)/log M. We also prove a corresponding statement for Ahlfors--David regular sets in [0,1]. The proof combines a multiscale entropy decomposition with inverse results from additive combinatorics, showing that Ahlfors--David regularity forces a definite entropy gain at each scale. The dependence 1/log M in the exponent is of optimal order.
Related
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- Decomposition of functions of bounded variation in Ahlfors-regular PI spaces
- Failure of dimension drop for harmonic measure on Ahlfors regular flat boundaries
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- Sidon sets with Δ-separated sumsets in additive number theory
BibTeX
@misc{tyrrell2026sumsets,
title = {{Sumsets of Ahlfors--David regular sets}},
author = {Fred Tyrrell},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.19299},
eprint = {2609.19299},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.19299},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Sumsets of Ahlfors--David regular sets
AU - Fred Tyrrell
PY - 2026
DA - 2026/09/16
PB - arXiv
DO - 10.48550/arxiv.2609.19299
UR - https://arxiv.org/abs/2609.19299
AB - We prove that there is an absolute constant c>0 such that for every finite (α,M)-Ahlfors--David regular set A⊆[N] with 0≤α<1 and M≥2, we have the sumset estimate |A+A|≥ |A|^1+c(1−α)/log M. We also prove a corresponding statement for Ahlfors--David regular sets in [0,1]. The proof combines a multiscale entropy decomposition with inverse results from additive combinatorics, showing that Ahlfors--David regularity forces a definite entropy gain at each scale. The dependence 1/log M in the exponent is of optimal order.
ER -