Borel reducibility and finitely Hölder(α) embeddability

Let (Xn,dn),n∈N, be a sequence of pseudo-metric spaces, and let p≥1. For x,y∈∏n∈NXn, let (x,y)∈E((Xn)n∈N;p)⇔∑n∈Ndn(x(n),y(n))p<+∞. For Borel reducibility between equivalence relations E((Xn)n∈N;p), we show it is closely related to finitely Hölder(α) embeddability between pseudo-metric spaces.

► We investigate Borel reducibility between lp-like equivalence relations. ► We introduce the notion of C-finitely Hölder(α) embeddability. ► We show the two notions are equivalent under widely applicable conditions.