Hausdorff measures of subgroups of mathbb{R }/mathbb{Z } and mathbb{R }

For a sequence $underline{u}=(u_n)_{nin mathbb{N }}$ of integers, let $t_{underline{u}}(mathbb{T })$ be the group of all topologically $underline{u}$ -torsion elements of the circle group $mathbb{T }:=mathbb{R }/mathbb{Z }$ . We show that for any $sin ]0,1[$ and $min {0,+infty }$ there exists $underline{u}$ such that $t_{underline{u}}(mathbb{T })$ has Hausdorff dimension $s$ and $s$ -dimensional Hausdorff measure equal to $m$ (no other values for $m$ are possible). More generally, for dimension functions $f,g$ with $f(t)prec g(t), f(t)prec !!!prec t$ and $g(t)prec !!!prec t$ we find $underline{u}$ such that $t_{underline{u}}(mathbb{T })$ has at the same time infinite $f$ -measure and null $g$ -measure.