Looking for examples of an uncountable proper subgroup of $(\mathbb R,+)$ without using the concept of Hamel basis of $\mathbb R$ over $\mathbb Q$ Please give some examples of an uncountable proper subgroup of $(\mathbb R,+)$ that does not depend on Hamel basis of $\mathbb R$ over $\mathbb Q$ . Using Hamel basis this is easy as we can find an injective endomorphism $f$ on $(\mathbb R,+)$ which is not a surjection , so $f(\mathbb R) (\ne \mathbb R)$ which has the same cardinaltiy as $\mathbb R$ , is a subgroup  of $(\mathbb R,+)$ , but I don't want to use the concept of Hamel basis , Please help 
 A: •  In [1], Erdös & Volkmann provide examples of additive subgroups $G$ of $\mathbb R$ for each Hausdorff dimension $s$ with $0 \le s \le 1$.  These are $F_\sigma$-sets. They have cardinal $2^{\aleph_0}$. Elements of the sets are those with certain "factorial base" expansions.  
•  A generalized construction is given in [2], Example 12.2:

Fix $0 < s < 1$.  Let $n_0, n_1, n_2,\dots$be a rapidly increasing sequence of integers, say with $n_{k+1} \ge \max\{n_k^k,4 n_k^{1/s}\}$ for each $k$.  For $r=1,2,3,\dots$ let
  $$
F_r = \{x \in \mathbb R\;:\; |x-p/n_k| \le r n_k^{-1/s} \text{ for some integer
$p$, for all $k$}\}
$$
  and let $F = \bigcup_{r=1}^\infty F_r$.  Then $\mathrm{dim}_{\mathrm{H}} F = s$,
  and $F$ is a subgroup of $\mathbb R$ under addition.  

• In [3] Chris Miller and I showed that a proper Borel  subring of $\mathbb R$ can only have Hausdorff dimension $0$.  But it may still be uncountable.  If $C \subseteq \mathbb R$ is a Cantor set such that all Cartesian powers $C^k$ have Hausdorff dimension $0$, then the ring $\mathbb Z[C]$ generated by it is uncountable and has Hausdorff dimension $0$.
[1] P. Erdös & B. Volkmann, "Additive Gruppen mit vorgegebener Hausdorffscher Dimension", J. Reine Angew. Math. 221 (1966) 203--208
[2] K. Falconer, Fractal Geometry: Mathematical Foundations and Applications, Wiley, 2003
[3] G. Edgar & C. Miller, "Borel subrings of the reals", Proc. Amer. Math. Soc.
131 (2002) 1121--1129
