Words and Transcendence

Abstract : Is it possible to distinguish algebraic from transcendental real numbers by considering the $b$-ary expansion in some base $b\ge2$? In 1950, É. Borel suggested that the answer is no and that for any real irrational algebraic number $x$ and for any base $g\ge2$, the $g$-ary expansion of $x$ should satisfy some of the laws that are shared by almost all numbers. For instance, the frequency where a given finite sequence of digits occurs should depend only on the base and on the length of the sequence. We are very far from such a goal: there is no explicitly known example of a triple $(g,a,x)$, where $g\ge3$ is an integer, $a$ a digit in $\{0,\ldots,g-1\}$ and $x$ a real irrational algebraic number, for which one can claim that the digit $a$ occurs infinitely often in the $g$-ary expansion of~$x$. Hence there is a huge gap between the established theory and the expected state of the art. However, some progress has been made recently, thanks mainly to clever use of Schmidt's subspace theorem. We review some of these results.
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Michel Waldschmidt. Words and Transcendence. W.W.L. Chen, W.T. Gowers, H. Halbertstam, W.M. Schmidt and R.C. Vaughan. Analytic Number Theory, Essays in Honour of Klaus Roth, Cambridge University Press, pp.449-470, 2009. ⟨hal-00407221⟩

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