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Vesselin Dimitrov
Vesselin Dimitrov
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This result is due originally to K. Mahler, and holds true more generally with any algebraic $a$ having $|a| > 1$ (so that the series converges absolutely). I can recommend Masser's lecture in the CIME 2000 school on diophantine approximations (LNM 1819), where the main idea of Mahler's proof is outlined as an illustration of the typical transcendence proof. The complete argument can be found in the opening chapter (Theorem 1.1.2) of K. Nishioka's book "Mahler Funtions and Transcendence" (LNM 1631), where you may also find various related results and generalizations.

Mahler's proof of the transcendence of $f(a^{-1})$ is based on the functional equation $f(z^b) = f(z) - z$ of the series $f(z) := \sum_{n \geq 1} z^{b^n}$. A completely different approach, which is based on Schmidt's Subspace theorem and allows for much more general transcendence statements, was discovered by P. Corvaja and U. Zannier in their article "Some new applications of the subspace theorem" (Compositio math, 2002).

This result is due originally to K. Mahler, and holds true more generally with any algebraic $a$ having $|a| > 1$ (so that the series converges absolutely). I can recommend Masser's lecture in the CIME 2000 school on diophantine approximations (LNM 1819), where the main idea of Mahler's proof is outlined as an illustration of the typical transcendence proof. The complete argument can be found in the opening chapter (Theorem 1.1.2) of K. Nishioka's book "Mahler Funtions and Transcendence" (LNM 1631), where you may also find various related results and generalizations.

A completely different approach, which is based on Schmidt's Subspace theorem and allows for much more general transcendence statements, was discovered by P. Corvaja and U. Zannier in their article "Some new applications of the subspace theorem" (Compositio math, 2002).

This result is due originally to K. Mahler, and holds true more generally with any algebraic $a$ having $|a| > 1$ (so that the series converges absolutely). I can recommend Masser's lecture in the CIME 2000 school on diophantine approximations (LNM 1819), where the main idea of Mahler's proof is outlined as an illustration of the typical transcendence proof. The complete argument can be found in the opening chapter (Theorem 1.1.2) of K. Nishioka's book "Mahler Funtions and Transcendence" (LNM 1631), where you may also find various related results and generalizations.

Mahler's proof of the transcendence of $f(a^{-1})$ is based on the functional equation $f(z^b) = f(z) - z$ of the series $f(z) := \sum_{n \geq 1} z^{b^n}$. A different approach, which is based on Schmidt's Subspace theorem and allows for much more general transcendence statements, was discovered by P. Corvaja and U. Zannier in their article "Some new applications of the subspace theorem" (Compositio math, 2002).

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Vesselin Dimitrov
Vesselin Dimitrov
  • 13.8k
  • 3
  • 56
  • 95

This result is due originally to K. Mahler, and holds true more generally with any algebraic $a$ having $|a| > 1$ (so that the series converges absolutely). I can recommend Masser's lecture in the CIME 2000 school on diophantine approximations (LNM 1819), where the main idea of Mahler's proof is outlined as an illustration of the typical transcendence proof. The complete argument can be found in the opening chapter (Theorem 1.1.2) of K. Nishioka's book "Mahler Funtions and Transcendence" (LNM 1631), where you may also find various related results and generalizations.

A completely different approach, which is based on Schmidt's Subspace theorem and allows for much more general transcendence statements, was discovered by P. Corvaja and U. Zannier in their article "Some new applications of the subspace theorem" (Compositio math, 2002).