The stabilized set of $p$'s in Krivine's theorem can be disconnected

TL;DR

This paper constructs a reflexive Banach space with a 1-unconditional basis where the set of $p$ for which $\, ext{ell}_p$ is finitely block represented can be any finite or certain countably infinite set, answering a longstanding question.

Contribution

It demonstrates that the stabilized Krivine set for a Banach space need not be connected, providing a new example with a prescribed set of $p$ values.

Findings

Constructed a reflexive Banach space with prescribed Krivine set.

Showed the Krivine set can be disconnected.

Identified the set of $p$ for spreading models in subspaces.

Abstract

For any closed subset $F$ of $[1,\infty]$ which is either finite or consists of the elements of an increasing sequence and its limit, a reflexive Banach space $X$ with a 1-unconditional basis is constructed so that in each block subspace $Y$ of $X$, $\ell_p$ is finitely block represented in $Y$ if and only if $p \in F$. In particular, this solves the question as to whether the stabilized Krivine set for a Banach space had to be connected. We also prove that for every infinite dimensional subspace $Y$ of $X$ there is a dense subset $G$ of $F$ such that the spreading models admitted by $Y$ are exactly the $\ell_p$ for $p\in G$.