**normal** $\; \not\Rightarrow \;$ **irrationality measure** $2$

There exist numbers that are normal with irrationality measure $>2$. In fact, there exist normal numbers (meaning normal with respect to every base) that have irrationality measure $\infty.$ This is Theorem 2 in Bugeaud **[1]** (2002). For related results, see **[2]** and **[3]**.

**irrationality measure** $2$ $\; \not\Rightarrow \;$ **normal**

There exist numbers with irrationality measure $2$ that are not normal. In fact, there exist numbers with irrationality measure $2$ that fail to be simply normal in every base. Note that this is stronger than failing to be simply normal in base $10,$ which in turn is stronger than failing to be normal in base $10,$ which in turn is stronger than "normal" in the sense that you are asking about. Shallit **[4]** (1979) showed that the continued fraction expansion of

$$\sum_{n=0}^\infty 2^{-2^{n}}$$

has bounded partial quotients (see Theorem 3 on p. 213 for meaning of $B(u,\infty)$ and Theorem 9 on p. 216 for the result), and thus has irrationality measure $2.$ [Indeed, for this we only need the $n$th partial quotient to be bounded by a linear function of $n.$ See Robert Israel's answer to the math overflow question Numbers with known irrationality measures.] However, it is clear that this number is not simply normal in any base. Indeed, in this number's expansion in any base, the proportion of $0$'s approaches $1,$ and hence the proportions for each of the other digits approaches $0.$

**[1]** Yann Bugeaud, *Nombres de Liouville et nombres normaux* [Liouville numbers and normal numbers], **Comptes Rendus Hebdomadaires des Séances de l'Académie des Sciences** 335 #2 (2002), 117-120.

**[2]** Verónica Becher, Pablo Heiber and Theodore A. Slaman, *A computable absolutely normal Liouville number*, preprint, 30 January 2014, 14 pages.

**[3]** Satyadev Nandakumar and Santhosh Kumar Vangapelli, *Normality and finite-state dimension of Liouville numbers*, arXiv:1204.4104v2, 21 January 2014.

**[4]** Jeffrey Outlaw Shallit, *Simple continued fractions for some irrational numbers*, **Journal of Number Theory** 11 #2 (April 1979), 209-217.

*(UPDATE, 41 MONTHS LATER)* A few days ago I happened to come across **[5]** below, which might be of interest to those finding this web page from an internet search for some known relations between normal numbers and Liouville numbers.

**[5]** Richard George Stoneham, *A general arithmetic construction of transcendental non-Liouville normal numbers from rational fractions*, **Acta Arithmetica** 16 #3 (1970), 239-253.