Explicit Witnesses at Every Gap of the Depth Filtration of βℕ
Abstract
Let Σ1 = ℕ* and Σk+1 = ℕ* + Σk̅ be the cumulative depth filtration of βℕ, the analogue for (ℕ,+) of a chain of closed ideals that Protasov and Protasova studied for discrete groups, where strict descent follows from a theorem of Lutsenko and Protasov. For every k we give an explicit set whose closure meets Σk but not Σk+1. Fix the doubly exponential sequence en = 22n, partition it into k subsequences E0, …, Ek−1 by the residue of the index modulo k, and set Ak = E0 + ⋯ + Ek−1. We prove that any sum q0 + ⋯ + qk−1 of free ultrafilters with Et ∈ qt lies in Σk ∖ Σk+1. The engine is a master lemma, proved by induction on j: if a sum F1 + ⋯ + Fj of subsequences of {en} with pairwise disjoint index sets belongs to a free ultrafilter s, then s ∉ Σj+1. The proof rests on a single rigidity of the doubly exponential sequence: a fixed difference forces the largest index in any shift-intersection, once it is large, to cancel within its own subsequence, which makes every shift-intersection descend by at least one level. The same witnesses lie in the gaps of the pure filtration.
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BibTeX
@misc{aza2026explicit,
title = {{Explicit Witnesses at Every Gap of the Depth Filtration of $β\mathbbN$}},
author = {Carl Aza},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.38696},
eprint = {2609.38696},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.38696},
}
RIS
TY - UNPB
TI - Explicit Witnesses at Every Gap of the Depth Filtration of βℕ
AU - Carl Aza
PY - 2026
DA - 2026/09/30
PB - arXiv
DO - 10.48550/arxiv.2609.38696
UR - https://arxiv.org/abs/2609.38696
AB - Let Σ_1 = ℕ^* and Σ_k+1 = ℕ^* + Σ_k̅ be the cumulative depth filtration of βℕ, the analogue for (ℕ,+) of a chain of closed ideals that Protasov and Protasova studied for discrete groups, where strict descent follows from a theorem of Lutsenko and Protasov. For every k we give an explicit set whose closure meets Σ_k but not Σ_k+1. Fix the doubly exponential sequence e_n = 2^2^n, partition it into k subsequences E_0, …, E_k−1 by the residue of the index modulo k, and set A_k = E_0 + ⋯ + E_k−1. We prove that any sum q_0 + ⋯ + q_k−1 of free ultrafilters with E_t ∈ q_t lies in Σ_k ∖ Σ_k+1. The engine is a master lemma, proved by induction on j: if a sum F_1 + ⋯ + F_j of subsequences of {e_n} with pairwise disjoint index sets belongs to a free ultrafilter s, then s ∉ Σ_j+1. The proof rests on a single rigidity of the doubly exponential sequence: a fixed difference forces the largest index in any shift-intersection, once it is large, to cancel within its own subsequence, which makes every shift-intersection descend by at least one level. The same witnesses lie in the gaps of the pure filtration.
ER -