Sharpening in the binomial case and an application

In our binomial case we have $b=1/k,\ c=1$, where $k\ge 3$ is a natural number. For a given $r\in\mathbb{Z}$, let us denote the denominators of

$$r/k\qquad\text{and}\qquad r/(k\underset{p \vert k}\prod p)$$

by k_* and k_**, respectively. We assume that $\hat D_n$ is a common factor of the numbers

$$\binom {n+\frac 1k}i\binom {n-\frac 1k}{n-i},\qquad i=0,\dots , n$$

satisfying $\hat D_n\ge e^{n\sigma(n_0,k)}$ for all $n\ge n_0$. Applying our Divisibility Criterion we show in paper II that there exists a sequence $\hat D_n$ with an asymptotic $\hat D_n\sim e^{n\sigma(\infty,k)}$, where

$$\sigma(\infty,k)=\frac 2{\phi(k)}\underset{(q,k)=1}{\sum_{k/2<q<k}}\left(\Psi(1)-\Psi\left(\frac qk\right)\right)$$

(here $\Psi$ means the digamma function). In paper III we also derive a formula for $\sigma(n_0,k)$ with $n_0\in \mathbb{Z} _+$.

For $n_0\in\mathbb{Z} _+\cup\{\infty\}$ we define

$$\hat Q(n_0,k)=\left(\sqrt s+\sqrt{s-r}\right)^2e^{-\sigma(n_0,k)} \min \{k_*\mu_k,k_{**}\},$$

$$\hat R(n_0,k)=\left(\sqrt s-\sqrt{s-r}\right)^2e^{-\sigma(n_0,k)} \min \{k_*\mu_k,k_{**}\}.$$

Theorem 5   If $k\ge 3$ is a natural number and r/s satisfies $\hat R(n_0,k)<1$ for some $n_0\in\mathbb{Z} _+\cup\{\infty\}$, then

$$m\left((1-r/s)^{-1/k}\right) \le 1-\frac {\ln \hat Q(n_0,k)}{\ln \hat R(n_0,k)},$$

where m means $m_{\text{asymp}}$ if $n_0=\infty$, and $m_{\text{eff}}$ if $n_0\in \mathbb{Z} _+$. In the latter case the constants c and N₀ are explicitly specified for every r/s, k and $n_0\in \mathbb{Z} _+$ satisfying the conditions of the theorem.

Paper III gives an extensive list of effective irrationality measures and corresponding constants for numbers of the form $\root k\of D$. This involves finding an appropriate solution to the Diophantine equation x^k-Dy^k=K, i.e. a solution such that K is small with respect to x and y. We show in paper III that solutions to this equation that exceed a certain bound are convergents of the continued fraction expansion of $\root k\of D$, and employ this fact in a systematic search for appropriate solutions, finding some cases not explicitly considered in preceding papers on this subject. Direct use of Theorem 5 gives results with N₀ quite large. Easton [15] has presented by means of an example a method involving the continued fractions to eliminate the bound N₀. We provide a general formulation of this method in paper III and employ it to obtain results that are true from N₀=0 on. Increasing n₀ in Theorem 5 has the effect of improving the measure, but the bound N₀ grows at the same time and the method involves computation of the continued fraction expansion up to N₀. Thus we should choose n₀ in such a way that we obtain as good a measure as possible, while N₀ remains within a range where we can apply the method described above. The new algorithm of Shiu [34] for the calculation of the continued fractions of algebraic numbers made it possible for us to allow the bound N₀ to be of the order of magnitude of 10²⁰⁰⁰⁰.

These results can be applied to the solution of the Diophantine equation

ax^k-by^k=K, (9)

usually called the Thue equation. More exactly, if we have a non-trivial irrationality measure m for $\root k\of {a/b}$ with an explicit constant c, we obtain an upper bound for the solutions x,y in terms of a, b, k, K, m and c. Some examples of bounds and solutions are also given. For instance, our results give the upper bound $1.3\times 10^{875}$ for the solutions of $\vert x^3-5 y^3\vert\le 100$ .

In fact, the history of the approximation of the binomial function is so closely connected with the problem of solving equation (4) that the two questions could not be handled separately. Thue [36], [37], [38], [39] was the first to deduce classical Padé approximations for the binomial function and to apply it to the solution of (4). Thue's ineffective improvement of the theorem of Liouville made it possible for him to restrict the number of solutions of (4) in certain cases. For later results concerning the number of solutions of (4) we refer the reader to [26] and [35].

The other line of investigation also originated by Thue is an attempt to determine upper bounds for the size of the solutions of (4). In fact, Thue's paper [40] includes results which are equivalent to non-trivial effective irrationality measures for the k:th roots of certain rationals, e.g. for $\root 3\of{(a+1)/a}$ with $a\ge 17$ and for $\root 7\of {17}$. Our method in paper III for the solution of (4) is essentially that of Thue refined in certain respects, especially by the use the common factor of the approximation polynomials as proposed by Chudnovsky [12].

It was almost half a century before Baker [2], [3] essentially rediscovered the method of Thue based on approximation of the hypergeometric function. Baker found, for example, that $m_{\text{eff}}(\root 3\of 2)\le 2.955$. Chudnovsky [12] was able to improve on the results of Baker, but he gives his measures only in asymptotic form, and these do not allow immediate effectivization, for this depends on effective knowledge of the distribution of primes in arithmetic progressions, and the polynomial bounds used are deduced from the Lemma of Poincaré. For example, he observed that $m_{\text{asymp}}(\root 3\of 2)\le 2.4298$. Chudnovsky also gave asymptotic bounds for some equations (4) with higher degrees, but without explicit proofs. The results of McCurley [25] made it possible for Easton [15], [16] to work out some cases handled by Chudnovsky in explicitly effective form. One example of his results is $m_{\text{eff}}(\root 3\of 2)\le 2.795$ with the constant $c=2.2\times 10^{-8}$ in (3).

A second method for computing irrationality measures for algebraic numbers or, equivalently, for determining upper bounds for the solutions of (4), is Baker's method involving linear forms of logarithms. With this approach Baker became the first to obtain a general effective improvement of the result of Liouville. First he [5] observed that the bound

$$\left\vert\vartheta-\frac MN\right\vert>cN^{-k}e^{(\log N)^{1/\kappa}},$$

where $\kappa>k+1$ and $c=c(\vartheta,\kappa)$, holds for every algebraic number $\vartheta$ of degree $k\ge 3$ and finally he [6] gave an improvement with a constant diminution of the Liouville bound. These results imply that the solutions of (4) are below a certain bound, and thus give a general algorithm for its solution. In many cases the algorithm works only in principle since the bounds are very large. All the improvements for special classes of algebraic numbers imply better bounds for solution of (4) in corresponding cases. This is perhaps the main ground for interest in this case. Baker and Stewart [8] also obtained using the linear forms of logarithms that $m(\root 3\of 5)\le 2.99999999999998$.

A third possible approach to this problem may be attributed to Bombieri and Mueller [9], who used their method involving the box principle to obtain a result for certain numbers of the form $\root k\of {a/b}$. If a and b are large numbers satisfying conditions of a certain type, their result compares favourably to those obtained by the hypergeometric method.

Our paper II generalizes the results of Chudnovsky [12], and agrees with them in all the examples given by him. In particular, we give an explicit formula for the common factor in both the asymptotic and the effective case for all $k\ge 3$. We also give the result $m_{\text{asymp}}(\root 3\of 5)\le 2.7636$ in paper III, which improves on that of Baker [8]. A systematic search with continued fractions also leads to some other new non-trivial measures, but there are still numbers for which the hypergeometric method seems not to give non-trivial measures, for example $\root 3\of {14}$.

In paper III we use the recent results of Ramaré and Rumely [28] on the distribution of primes, which are almost throughout sharper than those of McCurley [25]. On account of this and on our more accurate treatment of the common factor and related matters we are able to improve on the results of Easton for all the numbers presented in [15] and [16] and are also able to consider some cube roots not occuring in Easton's papers. As for higher roots of integers, our results provide the first explicitly presented irrationality measures in both asymptotic (paper II) and effective form (paper III).