## Rational approximations to algebraic numbers.(English)Zbl 0064.28501

Mathematika 2, 1-20 (1955); corrigendum 2, 168 (1955).
This important paper contains a proof of the long conjectured theorem:
“If $$\alpha$$ is an algebraic irrational number, and if there are infinitely many fractions $$h/q$$ with $$\vert \alpha - h/q\vert \le q^{-\kappa}$$, $$q > 0$$, $$(h, q) = 1$$, then $$\kappa\le 2$$.”
Much of the proof runs on classical lines. The new ideas are concerned with the multiple zeros at rational points of polynomials in many variables. If $$R(x_1,\ldots,x_m)\not\equiv 0$$ is such a polynomial, and if $$\alpha_1,\ldots, \alpha_m$$ are real numbers and $$r_1,\ldots, r_m$$ are positive integers, write
$R(\alpha_1+ y_1, \ldots, \alpha_m+ y_m) = \sum_{j_1=0}^\infty \cdots \sum_{j_m=0}^\infty c(j_1,\ldots, j_m) y_1^{j_1} \cdots y_m^{j_m}.$
Then the index of $$R$$ at $$(\alpha_1,\ldots, \alpha_m)$$ relative to $$r_1,\ldots, r_m$$ is defined as $\min (j_1/r_1 + \ldots + j_m/r_m)$ extended over all $$j_1,\ldots, j_m$$ with $$c(j_1,\ldots, j_m) \ne 0$$. This index is a non-Archimedean (logarithmic) valuation. Next let $$B$$, $$r_1,\ldots, r_m$$, $$q_1,\ldots, q_m$$ be positive integers, and let $$\mathfrak R_m = \mathfrak R_m(B; r_1,\ldots,r_m)$$ be the set of all polynomials
$R(x_1,\ldots,x_m) = \sum_{k_1=0}^{r_1}\cdots \sum_{k_m=0}^{r_m} a_{k_1\cdots k_m} x_1^{k_1} \cdots x_1^{k_1} \not\equiv 0$
with integral coefficients satisfying $$\vert a_{k_1\cdots k_m}\vert \le B$$. Denote by $$\Theta_m(B; q_1,\ldots,q_m; r_1,\ldots,r_m)$$ the upper bound of the index of $$R$$ at the point $$(h_1/q_1, \ldots, h_m/q_m)$$, extended over all $$R\in\mathfrak R_m$$ and over all integers $$h_1,\ldots, h_m$$ prime to $$q_1,\ldots, q_m$$, respectively. The main lemma states:
“Let $$m$$, $$r_1,\ldots, r_m$$ , $$q_1,\ldots, q_m$$ be positive integers and $$\delta$$ a real number such that $$0 < \delta <1/m$$, $$r_m > 10/\delta$$, $$r_{j-1}/r_j> 1/\delta$$ for $$j = 2, 3,\ldots, m$$, $$\log q_1 > m (2m + 1)/\delta$$, $$r_j\log q_j \ge r_1 \log q_1$$ for $$j = 2, 3,\ldots, m$$. Then
$\Theta_m\left(q_1^{\delta r_1}, q_1,\ldots, q_m; r_1,\ldots, r_m\right) < 10^m \delta^{(1/2)^m}.''$
This lemma is proved by induction for $$m$$.
If the theorem were false, one could select $$m$$ solutions $$h_j/q_j$$ of $$\vert \alpha - h_j/q_j\vert < q_j^{-\kappa}$$, where $$\kappa > 2$$, and construct a polynomial $$R\ne 0$$ with not too large integral coefficients, which would be (i) of high index at $$(\alpha,\ldots,\alpha)$$, but (ii) of low index at $$P = (h_1/q_1,\ldots,h_m/q_m)$$. A certain partial derivative of $$R$$, taken at $$P$$, would then, by (i), be very small in absolute value, while, by (ii), it would be $$\ne 0$$. As $$R$$ has a rational value of denominator $$q_1^{r_1}\cdots q_m^{r_m}$$, one would arrive at a contradiction.

### MSC:

 11J68 Approximation to algebraic numbers
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