Reducibility and nonreducibility between $\ell ^p$ equivalence relations

Abstract:

We show that, for $1 \le p < q < \infty$, the relation of $\ell ^{p}$-equivalence between infinite sequences of real numbers is Borel reducible to the relation of $\ell ^{q}$-equivalence (i.e., the Borel cardinality of the quotient ${\mathbb {R}}^{\mathbb {N}}/\ell ^{p}$ is no larger than that of ${\mathbb {R}}^{\mathbb {N}}/\ell ^{q}$), but not vice versa. The Borel reduction is constructed using variants of the triadic Koch snowflake curve; the nonreducibility in the other direction is proved by taking a putative Borel reduction, refining it to a reduction map that is not only continuous but ‘modular,’ and using this nicer map to derive a contradiction.