3 problems

Dimension bound conjecture. The dimension of a topological basis of a topologically free subgroup of a free abelian topological group $A(X)$ cannot exceed $\dim X$:

Let $W(G)$ denote the word problem of a group $G$ with respect to a finite symmetric generating set. In particular, $W(\mathbb{Z}^n)$ is the set of words over a finite generating s…

Let $F_{1,n,n}(r)$ denote the sum of the reciprocals of the absolute determinants of all full-rank $n\times n$ integer matrices whose entries have Euclidean norm at most $r$, and l…