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In this paper we introduce the notion of almost equality (or, more specifically, almost equality by translations) of complex functions of an unrestricted real variable in terms of the new concept of  ϵ -translation number of a function with respect to other one, which is inspired by Bohr’s notion of  ϵ -translation number associated with an almost periodic function. We develop the main properties of this new class of functions and obtain a characterization through a very important equivalence relation which we introduced in previous papers in the context of the almost periodicity.

Equivalent almost periodic functions in terms of the new property of almost equality