  Possible transcendental nature of Euler's constant $\gamma$ has been the
focus of study for sometime now. One possible approach is to consider $\gamma$
not in isolation, but as an element of the infinite family of generalised
Euler-Briggs constants. In a recent work \cite{GSS}, it is shown that the
infinite list of generalized Euler-Briggs constants can have at most one
algebraic number. In this paper, we study the dimension of spaces generated by
these generalized Euler-Briggs constants over number fields. More precisely, we
obtain non-trivial lower bounds (see \thmref{pre} and \thmref{linear-ind}) on
the dimension of these spaces and consequently establish the infinite
dimensionality of the space spanned. Further, we study linear and algebraic
independence of these constants over the field of all algebraic numbers.
