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I’ve been focusing my blog posts recently on the mathematics around Hilbert’s fifth problem (is every locally Euclidean group a Lie group?), but today, I will be discussing another of Hilbert’s problems, namely Hilbert’s seventh problem, on the transcendence of powers ${a^b}$ of two algebraic numbers ${a,b}$. (I am not randomly going through Hilbert’s list, by the way; I hope to explain my interest in the seventh problem in a later post.) This problem was famously solved by Gelfond and Schneider in the 1930s:

Theorem 1 (Gelfond-Schneider theorem) Let ${a, b}$ be algebraic numbers, with ${a \neq 0,1}$ and ${b}$ irrational. Then (any of the values of the possibly multi-valued expression) ${a^b}$ is transcendental.

For sake of simplifying the discussion, let us focus on just one specific consequence of this theorem:

Corollary 2 ${\frac{\log 2}{\log 3}}$ is transcendental.

Proof: If not, one could obtain a contradiction to the Gelfond-Schneider theorem by setting ${a := 3}$ and ${b := \frac{\log 2}{\log 3}}$. (Note that ${\frac{\log 2}{\log 3}}$ is clearly irrational, since ${3^p \neq 2^q}$ for any integers ${p,q}$ with ${q}$ positive.) $\Box$

In the 1960s, Alan Baker established a major generalisation of the Gelfond-Schneider theorem known as Baker’s theorem, as part of his work in transcendence theory that later earned him a Fields Medal. Among other things, this theorem provided explicit quantitative bounds on exactly how transcendental quantities such as ${\frac{\log 2}{\log 3}}$ were. In particular, it gave a strong bound on how irrational such quantities were (i.e. how easily they were approximable by rationals). Here, in particular, is one special case of Baker’s theorem:

Proposition 3 (Special case of Baker’s theorem) For any integers ${p, q}$ with ${q}$ positive, one has

$\displaystyle |\frac{\log 2}{\log 3} - \frac{p}{q}| \geq c \frac{1}{q^C}$

for some absolute (and effectively computable) constants ${c, C > 0}$.

This theorem may be compared with (the easily proved) Liouville’s theorem on diophantine approximation, which asserts that if ${\alpha}$ is an irrational algebraic number of degree ${d}$, then

$\displaystyle |\alpha - \frac{p}{q}| \geq c \frac{1}{q^d}$

for all ${p,q}$ with ${q}$ positive, and some effectively computable ${c>0}$, and (the more significantly difficult) Thue-Siegel-Roth theorem, which under the same hypotheses gives the bound

$\displaystyle |\alpha - \frac{p}{q}| \geq c_\epsilon \frac{1}{q^{2+\epsilon}}$

for all ${\epsilon>0}$, all ${p,q}$ with ${q}$ positive and an ineffective constant ${c_\epsilon>0}$. Finally, one should compare these results against Dirichlet’s theorem on Diophantine approximation, which asserts that for any real number ${\alpha}$ one has

$\displaystyle |\alpha - \frac{p}{q}| < \frac{1}{q^2}$

for infinitely many ${p,q}$ with ${q}$ positive. (The reason the Thue-Siegel-Roth theorem is ineffective is because it relies heavily on the dueling conspiracies argument, i.e. playing off multiple “conspiracies” ${\alpha \approx \frac{p}{q}}$ against each other; the other results however only focus on one approximation at a time and thus avoid ineffectivity.)

Proposition 3 easily implies the following separation property between powers of ${2}$ and powers of ${3}$:

Corollary 4 (Separation between powers of ${2}$ and powers of ${3}$) For any positive integers ${p, q}$ one has

$\displaystyle |3^p - 2^q| \geq \frac{c}{q^C} 3^p$

for some effectively computable constants ${c, C > 0}$ (which may be slightly different from those in Proposition 3).

Indeed, this follows quickly from Proposition 3, the identity

$\displaystyle 3^p - 2^q = 3^p ( 1 - 3^{q (\frac{\log 2}{\log 3} - \frac{p}{q})}) \ \ \ \ \ (1)$

and some elementary estimates.

In particular, the gap between powers of three ${3^p}$ and powers of two ${2^q}$ grows exponentially in the exponents ${p,q}$. I do not know of any other way to establish this fact other than essentially going through some version of Baker’s argument (which will be given below).

For comparison, by exploiting the trivial (yet fundamental) integrality gap – the obvious fact that if an integer ${n}$ is non-zero, then its magnitude is at least ${1}$ – we have the trivial bound

$\displaystyle |3^p - 2^q| \geq 1$

for all positive integers ${p, q}$ (since, from the fundamental theorem of arithmetic, ${3^p-2^q}$ cannot vanish). Putting this into (1) we obtain a very weak version of Proposition 3, that only gives exponential bounds instead of polynomial ones:

Proposition 5 (Trivial bound) For any integers ${p, q}$ with ${q}$ positive, one has

$\displaystyle |\frac{\log 2}{\log 3} - \frac{p}{q}| \geq c \frac{1}{2^q}$

for some absolute (and effectively computable) constant ${c > 0}$.

The proof of Baker’s theorem (or even of the simpler special case in Proposition 3) is largely elementary (except for some very basic complex analysis), but is quite intricate and lengthy, as a lot of careful book-keeping is necessary in order to get a bound as strong as that in Proposition 3. To illustrate the main ideas, I will prove a bound that is weaker than Proposition 3, but still significantly stronger than Proposition 5, and whose proof already captures many of the key ideas of Baker:

Proposition 6 (Weak special case of Baker’s theorem) For any integers ${p, q}$ with ${q > 1}$, one has

$\displaystyle |\frac{\log 2}{\log 3} - \frac{p}{q}| \geq \exp( - C \log^{C'} q )$

for some absolute constants ${C, C' > 0}$.

Note that Proposition 3 is equivalent to the assertion that one can take ${C'=1}$ (and ${C}$ effective) in the above proposition.

The proof of Proposition 6 can be made effective (for instance, it is not too difficult to make the ${C'}$ close to ${2}$); however, in order to simplify the exposition (and in particular, to use some nonstandard analysis terminology to reduce the epsilon management), I will establish Proposition 6 with ineffective constants ${C, C'}$.

Like many other results in transcendence theory, the proof of Baker’s theorem (and Proposition 6) rely on what we would nowadays call the polynomial method – to play off upper and lower bounds on the complexity of polynomials that vanish (or nearly vanish) to high order on a specified set of points. (I have discussed the polynomial method in relation to problems in incidence geometry in several previous blog posts.) In the specific case of Proposition 6, the points in question are of the form

$\displaystyle \Gamma_N := \{ (2^n, 3^n): n = 1,\ldots,N \} \subset {\bf R}^2$

for some large integer ${N}$. On the one hand, the irrationality of ${\frac{\log 2}{\log 3}}$ ensures that the curve

$\displaystyle \gamma := \{ (2^t, 3^t): t \in {\bf R} \}$

is not algebraic, and so it is difficult for a polynomial ${P}$ of controlled complexity to vanish (or nearly vanish) to high order at all the points of ${\Gamma_N}$; the trivial bound in Proposition 5 allows one to make this statement more precise. (Here, “complexity” of a polynomial is an informal term referring both to the degree of the polynomial, and the height of the coefficients, which in our application will essentially be integers up to some normalisation factors.) On the other hand, if Proposition 6 failed, then ${\frac{\log 2}{\log 3}}$ is close to a rational, which by Taylor expansion makes ${\gamma}$ close to an algebraic curve over the rationals (up to some rescaling by factors such as ${\log 2}$ and ${\log 3}$) at each point of ${\Gamma_N}$. This, together with a pigeonholing argument, allows one to find a polynomial ${P}$ of reasonably controlled complexity to (nearly) vanish to high order at every point of ${\Gamma_N}$.

These observations, by themselves, are not sufficient to get beyond the trivial bound in Proposition 5. However, Baker’s key insight was to exploit the integrality gap to bootstrap the (near) vanishing of ${P}$ on a set ${\Gamma_N}$ to imply near-vanishing of ${P}$ on a larger set ${\Gamma_{N'}}$ with ${N'>N}$. The point is that if a polynomial ${P}$ of controlled degree and size (nearly) vanishes to higher order on a lot of points on an analytic curve such as ${\gamma}$, then it will also be fairly small on many other points in ${\gamma}$ as well. (To quantify this statement efficiently, it is convenient to use the tools of complex analysis, which are particularly well suited to understand zeroes (or small values) of polynomials.) But then, thanks to the integrality gap (and the controlled complexity of ${P}$), we can amplify “fairly small” to “very small”.

Using this observation and an iteration argument, Baker was able to take a polynomial of controlled complexity ${P}$ that nearly vanished to high order on a relatively small set ${\Gamma_{N_0}}$, and bootstrap that to show near-vanishing on a much larger set ${\Gamma_{N_k}}$. This bootstrap allows one to dramatically bridge the gap between the upper and lower bounds on the complexity of polynomials that nearly vanish to a specified order on a given ${\Gamma_N}$, and eventually leads to Proposition 6 (and, with much more care and effort, to Proposition 3).

Below the fold, I give the details of this argument. My treatment here is inspired by this expose of Serre, and these lecture notes of Soundararajan (as transcribed by Ian Petrow).