The Jónsson distributivity spectrum

Suppose throughout that V is a congruence distributive variety. If m ≥ 1, let J V (m) be the smallest natural number k such that the congruence identity α(β ◦ γ ◦ β . . . ) ⊆ αβ ◦ αγ ◦ αβ ◦ . . . holds in V , with m occurrences of ◦ on the left and k occurrences of ◦ on the right. We show that if J V (m) = k, then J V (m) ≤ k, for every natural number . If J V (1) = 2, that is, V is 3-distributive, then J V (m) ≤ m, for every m ≥ 3. If V is m-modular, that is, congruence modularity of V is witnessed by m + 1 Day terms, then J V (2) ≤ J V (1) + 2m 2 − 2m − 1. Open problems are stated at various places.