On simultaneous approximations of two algebraic numbers by rationals. (English) Zbl 0173.04801

Schmidt, Wolfgang M.

Acta Math. 119, 27-50 (1967).

The author establishes a well-known conjecture, extending the famous theorem of K. F. Roth [Mathematika 2, 1–20 (1955; Zbl 0064.28501)]. He proves that if \(\alpha, \beta\) are algebraic numbers such that \(1, \alpha, \beta\) are linearly independent over the rationals then, for any \(\varepsilon > 0\), there exist only finitely many positive integers \(q\) such that \[ \Vert q \alpha\Vert \, \Vert q \beta\Vert q^{1+\varepsilon} < 1 \tag{*} \] – here, as usual, \(\Vert x\Vert\) denotes the distance of \(x\) from the nearest integer. The theorem implies, in particular, that there exist only finitely many pairs of rationals \(p_1/q, p_2/q\) such that \[ \left|\alpha - p_1/q \right| < |q|^{-3/2 - \varepsilon}, \quad \left|\beta - p_2/q \right| < |q|^{-3/2 - \varepsilon}, \] and, dually, that there exist only finitely many integers \(p, q_1, q_2\) such that \[ \left|q_1 \alpha + q_2 \beta + p\right|<q^{-2 - \varepsilon}, \] where \(q=\max \left(\left|q_1\right|, \left|q_2\right|\right)>0 \). This further implies a best-possible result on the approximation of algebraic numbers by rationals or quadratic irrationals. The structure of the proof is similar to that of the author’s earlier work [Acta Math. 102, 159–224 (1959; Zbl 0215.35104)].
It is shown first, by techniques from the geometry of numbers, that (*) implies the existence of a pair \(\mathfrak{m}_1, \mathfrak{w}_2\) of independent integer triplets satisfying one or other of a set of three linear inequalities with mutually adjoint forms. Let either of these sets of forms be denoted by \(L_1, L_2, L_3\). It is assumed that (*) has infinitely many solutions and \(m\) are selected so that the corresponding pairs \(\mathfrak{w}_{h 1}, \mathfrak{w}_{h 2}, (1 \leq h\leq m)\) satisfy conditions of magnitude of the type required by Roth.
Let \(M_h\) denote the unique linear form with relatively prime integer coefficients which vanishes at \(\mathfrak{w}_{h 1}, \mathfrak{w}_{h 2} \). A polynomial \( P\left(X_{11}, X_{12}, X_{13}, \ldots, X_{m 1}, X_{m 2}, X_{m 3}\right) \) in \( 3 m \) variables is constructed which is homogeneous in \( X_{h 1}, X_{h 2}, X_{h 3}(1 \leq h \leq m) \) and which vanishes to a high degree with respect to \( L_1, L_2, L_3\): By a modified version of Roth’s lemma it is shown that the index of \(P\) with respect to \(M_1, \ldots, M_m\) is small. On the other hand, the construction implies that \(P\) and its derivatives to a high order must vanish at all points of a certain “grid” based on the \( \mathfrak{w}_{h 1}, \mathfrak{w}_{h 2} \), whence \(P\) must have a large index with respect to \(M_1, \ldots, M_m\). The contradiction proves the theorem.
The main innovation as compared with the author’s earlier work concerns the utilization of the derivatives of \(P\) in the latter argument, thereby allowing a contraction in the size of the grid.

Reviewer: Alan Baker (Cambridge)

MSC:

11J13	Simultaneous homogeneous approximation, linear forms
11J68	Approximation to algebraic numbers

Keywords:

simultaneous approximations; two algebraic numbers by rationals

Citations:

Zbl 0064.28501; Zbl 0215.35104

References:

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