Realising sets of integers as mapping degree sets

Given two closed oriented manifolds M,N$M,N$ of the same dimension, we denote the set of degrees of maps from M$M$ to N$N$ by D(M,N)$D(M,N)$. The set D(M,N)$D(M,N)$ always contains zero. We show the following (non-)realisability results. There exists an infinite subset A$A$ of Z$\mathbb {Z}$ containing 0 which cannot be realised as D(M,N)$D(M,N)$, for any closed oriented n$n$-manifolds M,N$M,N$.Every finite arithmetic progression of integers containing 0 can be realised as D(M,N)$D(M,N)$, for some closed oriented 3-manifolds M,N$M,N$.Together with 0, every finite geometric progression of positive integers starting from 1 can be realised as D(M,N)$D(M,N)$, for some closed oriented manifolds M,N$M,N$.(i)(ii)(iii)