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In 1984, K. Mahler asked how well elements in the Cantor middle third set can be approximated by rational numbers from that set and by rational numbers outside of that set. We consider more general missing digit sets 
$C$ and construct numbers in 
$C$ that are arbitrarily well approximable by rationals in 
$C$, but badly approximable by rationals outside of 
$C$. More precisely, we construct them so that all but finitely many of their convergents lie in 
$C$.
Fix an integer 
$n\geqslant 2$ . To each non-zero point 
$\mathbf{u}$ in 
$\mathbb{R}^{n}$ , one attaches several numbers called exponents of Diophantine approximation. However, as Khintchine first observed, these numbers are not independent of each other. This raises the problem of describing the set of all possible values that a given family of exponents can take by varying the point 
$\mathbf{u}$ . To avoid trivialities, one restricts to points 
$\mathbf{u}$ whose coordinates are linearly independent over 
$\mathbb{Q}$ . The resulting set of values is called the spectrum of these exponents. We show that, in an appropriate setting, any such spectrum is a compact connected set. In the case 
$n=3$ , we prove moreover that it is a semi-algebraic set closed under component-wise minimum. For 
$n=3$ , we also obtain a description of the spectrum of the exponents 
$(\text{}\underline{\unicode[STIX]{x1D711}}_{1},\text{}\underline{\unicode[STIX]{x1D711}}_{2},\text{}\underline{\unicode[STIX]{x1D711}}_{3},\overline{\unicode[STIX]{x1D711}}_{1},\overline{\unicode[STIX]{x1D711}}_{2},\overline{\unicode[STIX]{x1D711}}_{3})$ recently introduced by Schmidt and Summerer.
$\mathbb {G}_{{\textrm{a}}}\times \mathbb {G}_{{\textrm{m}}}$
A small value estimate is a statement providing necessary conditions for the existence of certain sequences of non-zero polynomials with integer coefficients taking small values at points of an algebraic group. Such statements are desirable for applications to transcendental number theory to analyze the outcome of the construction of an auxiliary function. In this paper, we present a result of this type for the product 
$ \mathbb {G}_{\mathrm {a}}\times \mathbb {G}_{\mathrm {m}}$ whose underlying group of complex points is 
$\mathbb {C}\times \mathbb {C}^{*}$. It shows that if a certain sequence of non-zero polynomials in 
$ \mathbb {Z}[X_1,X_2]$ takes small values at a point 
$(\xi ,\eta )$ together with their first derivatives with respect to the invariant derivation 
$\partial /\partial X_1 + X_2 (\partial /\partial X_2)$, then both 
$\xi $ and 
$\eta $ are algebraic over 
$\mathbb {Q}$. The precise statement involves growth conditions on the degree and norm of these polynomials as well as on the absolute values of their derivatives. It improves on a direct application of Philippon’s criterion for algebraic independence and compares favorably with constructions coming from Dirichlet’s box principle.
For each real number 
$\xi$ , let 
$\widehat{{{\lambda }_{2}}}\left( \xi\right)$ denote the supremum of all real numbers 
$\text{ }\!\!\lambda\!\!\text{ }$ such that, for each sufficiently large 
$X$ , the inequalities 
$\left| {{x}_{0}} \right|\,\le \,X,\,\left| {{x}_{0}}\xi \,-\,{{x}_{1}} \right|\,\le \,{{X}^{-\lambda \text{ }}}$ and 
$\left| {{x}_{0}}{{\xi }^{2}}\,-\,{{x}_{2}} \right|\,\le \,{{X}^{-\lambda \text{ }}}$ admit a solution in integers 
${{x}_{0}},\,{{x}_{1}}$ and 
${{x}_{2}}$ not all zero, and let 
$\widehat{{{\omega }_{2}}}\left( \xi\right)$ denote the supremum of all real numbers 
$\omega $ such that, for each sufficiently large $X$ , the dual inequalities 
$\left| {{x}_{0}}\,+\,{{x}_{1}}\xi \,+\,{{x}_{2}}{{\xi }^{2}} \right|\,\le \,{{X}^{-\omega }}$ , 
$\left| {{x}_{1}} \right|\,\le \,X$ and 
$\left| {{x}_{2}} \right|\,\le \,X$ admit a solution in integers ${{x}_{0}},\,{{x}_{1}}$ and ${{x}_{2}}$ not all zero. Answering a question of Y. Bugeaud and M. Laurent, we show that the exponents $\widehat{{{\lambda }_{2}}}\left( \xi\right)$ where 
$\xi$ ranges through all real numbers with 
$[\mathbb{Q}(\xi )\,:\mathbb{Q}]\,>\,2$ form a dense subset of the interval 
$\left[ 1/2,\,\left( \sqrt{5}\,-\,1 \right)/2 \right]$ while, for the same values of $\xi$ , the dual exponents 
$\widehat{{{\omega }_{2}}}\left( \xi\right)$ form a dense subset of 
$\left[ 2,\,\left( \sqrt{5}\,+\,3 \right)/2 \right]$ . Part of the proof rests on a result of V. Jarník showing that 
$\widehat{{{\lambda }_{2}}}\left( \xi\right)=1-{{\hat{\omega }}_{2}}{{\left( \xi\right)}^{-1}}$ for any real number $\xi$ with $[\mathbb{Q}(\xi )\,:\mathbb{Q}]\,>\,2$ .
The purpose of this paper is to show the limitations of the conjectures of algebraic approximation. For this, we construct points of 
${{\mathbf{C}}^{m}}$ which do not admit good algebraic approximations of bounded degree and height, when the bounds on the degree and the height are taken from specific sequences. The coordinates of these points are Liouville numbers.
