Bourgain's Lambda(p) selection theorem: a greedy proof with polynomial failure bounds
Abstract
We give a self-contained proof of Bourgain's finite Λ(p) selection theorem with an explicit probability bound. For every p>2 and N≥2, given N pairwise orthogonal functions bounded by one on a probability space, a uniformly chosen subsystem of cardinality ⌈ N2/p⌉ satisfies the Λ(p) inequality with probability at least 1−1/N, with a constant depending only on p. The inequality holds simultaneously for all complex coefficient vectors. More generally, for every fixed A>0, the success probability is at least 1−1/NA with a constant depending only on p and A, and independent of N. The proof uses a greedy approximation in Lr, with r>2, whose potential decreases throughout all approximation scales. A single weighted bound on the cumulative number of updates controls a union bound over discrete update histories, and increasing the weight assigned to each history gives the prescribed failure exponent without changing the cardinality exponent.
Related
- Bourgain's proof of the circular maximal theorem: an exposition
- Lower bounds for some value sets over finite fields: incidence geometry and Bourgain's group expansion theorem
- Bourgain--Brezis selection maps for Hodge systems in Besov and Triebel--Lizorkin spaces
- Further bounds in the polynomial Szemerédi theorem
- A functional central limit theorem for particle systems under polynomial growth conditions, with applications to statistics
BibTeX
@misc{burstein2026bourgains,
title = {{Bourgain's Lambda(p) selection theorem: a greedy proof with polynomial failure bounds}},
author = {Will Burstein and Alex Iosevich and Ben Krause},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.12566},
eprint = {2609.12566},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.12566},
}
RIS
TY - UNPB
TI - Bourgain's Lambda(p) selection theorem: a greedy proof with polynomial failure bounds
AU - Will Burstein
AU - Alex Iosevich
AU - Ben Krause
PY - 2026
DA - 2026/09/11
PB - arXiv
DO - 10.48550/arxiv.2609.12566
UR - https://arxiv.org/abs/2609.12566
AB - We give a self-contained proof of Bourgain's finite Λ(p) selection theorem with an explicit probability bound. For every p>2 and N≥2, given N pairwise orthogonal functions bounded by one on a probability space, a uniformly chosen subsystem of cardinality ⌈ N^2/p⌉ satisfies the Λ(p) inequality with probability at least 1−1/N, with a constant depending only on p. The inequality holds simultaneously for all complex coefficient vectors. More generally, for every fixed A>0, the success probability is at least 1−1/N^A with a constant depending only on p and A, and independent of N. The proof uses a greedy approximation in L^r, with r>2, whose potential decreases throughout all approximation scales. A single weighted bound on the cumulative number of updates controls a union bound over discrete update histories, and increasing the weight assigned to each history gives the prescribed failure exponent without changing the cardinality exponent.
ER -