Approximation of $k$-ary functions by functions from the given system

Problems of $k$-ary functions approximation by functions from the given system are investigated in this paper. In particular, generalization of Golomb theorem [1] is obtained in the case of ring $\mathbb{Z}/k$ or finite field $GF(q)$. The definition of $k$-ary functions equivalency with respect to the given functions system is introduced. Classes of equivalency with respect to the linear functions system over finite field or ring $\mathbb{Z}/4$ are described. Limit theorems on cardinality of random $k$-ary functions equivalency class are proved. Also in this paper we found functions which minimize maximum probability of coincidence with linear functions in one variable over finite ring with identity.