Effective irrationality measures for algebraic numbers: the unresolved questions of Bombieri's 1982 Thue-Siegel-Dyson method, toward an effective Roth's theorem

How well can an irrational algebraic number be approximated by rationals? Roth's theorem (1955) gives the optimal exponent: for every irrational algebraic $\alpha$ and every $\varepsilon > 0$ there is a constant $c(\alpha, \varepsilon) > 0$ with $|\alpha - p/q| > c\, H(p/q)^{-2-\varepsilon}$ for all rationals $p/q$, where $H$ is the multiplicative height. In every known proof this constant is not effectively computable - no algorithm recovers it from $\alpha$ and $\varepsilon$. This effectiveness gap is a central open problem of Diophantine approximation. Bombieri's 1982 refinement of the Thue-Siegel-Dyson method obtained effective irrationality exponents of Dyson's order $\sqrt{2r}$ ($r$ = degree) for large classes of numbers and isolated four concrete obstacles to pushing the exponent toward the optimal value 2: (1) sharpening the height bound for the auxiliary polynomial (Lemma 6 of that paper); (2) extending Dyson's two-variable zero estimate to several variables; (3) removing the rapidly-increasing-heights requirement from Roth's lemma; (4) constructing explicit anchor approximations of the extraordinary quality the effective method needs. Obstacle (2) was removed by Esnault and Viehweg (1984), though for the Mordell program rather than for effectivity; the others, and the effective-Roth objective itself, remain open as of 2026, with effective measures known only for restricted families and, most recently, conditional on an effective abc conjecture (a computable-constants version of the conjectural bound relating $a + b = c$ to the prime factors of $abc$).

Background

Rational approximation of algebraic numbers is organized by the irrationality exponent: $\mu(\alpha)$ is the supremum of $\mu$ such that $|\alpha - p/q| < q^{-\mu}$ holds for infinitely many rationals $p/q$ in lowest terms; here $H(p/q) = \max(|p|, q)$ is the (multiplicative) height and $h = \log H$ the logarithmic height, extended to algebraic numbers as the Weil height. Liouville's elementary bound gives $\mu(\alpha) \le r$ for $\alpha$ of degree $r \ge 3$. Successive refinements lowered this: Thue ($r/2 + 1$), Siegel ($2\sqrt{r}$), Dyson and independently Gelfond ($\sqrt{2r}$), and finally Roth ($2 + \varepsilon$, optimal); Schmidt's monograph (1980) gives a unified account of this development. A bound is called effective when its constants can be computed from the data (the degree and height of $\alpha$, and $\varepsilon$); Roth's bound is the sharpest but the least effective - like Thue's, Siegel's and Dyson's, its proof is a reductio that leaves the constant non-computable, whereas Baker's theory of linear forms in logarithms (effective lower bounds for nonzero expressions $\beta_1 \log \alpha_1 + \dots + \beta_n \log \alpha_n$ with algebraic $\alpha_i, \beta_i$; Baker 1966) is effective for all algebraic numbers but only improves the Liouville exponent $r$ by an amount tending to 0.

The common engine of the Thue-Siegel-Dyson-Roth family is an auxiliary polynomial: a polynomial $P(x_1, x_2)$ of bidegree at most $(d_1, d_2)$ (degree at most $d_1$ in $x_1$ and at most $d_2$ in $x_2$) constructed to vanish to high order at $(\alpha_1, \alpha_2)$, where $\alpha_1, \alpha_2$ are algebraic and one takes $\alpha_1 = \alpha_2 = \alpha$ to study approximations of $\alpha$. Dyson's lemma is the zero estimate that bounds the total order of vanishing such a polynomial can have at the finitely many points attached to the conjugates of the $\alpha_i$; combined with the product formula (the identity stating that the product of all normalized absolute values of a nonzero algebraic number equals 1) it yields the $\sqrt{2r}$ exponent. Bombieri's 1982 Acta Mathematica paper refined Dyson's lemma (his Theorem 1) so that it holds over the full range of vanishing orders (earlier forms required the orders to be balanced), and combined it with an explicit height bound for the auxiliary polynomial obtained from an application of Siegel's lemma - the principle that an underdetermined homogeneous linear system with integer coefficients admits a nonzero solution of explicitly bounded height - over number fields (his Lemma 6: a bound linear in the bidegrees $d_1, d_2$ weighted by height terms of $\alpha_1$ and $\alpha_2$, with an explicit coefficient depending on the degree $r$ and on auxiliary parameters of the construction). The result was a genuinely effective Thue-Siegel-Dyson theorem: effective exponents of $\sqrt{2r}$ scale for all generators of suitable number fields, where earlier effective methods gave essentially nothing beyond Liouville.

The effective branch of Bombieri's Theorem 4 is powered by what later literature calls an excellent approximation, or anchor. Theorem 4 is an alternative: either a power-law lower bound $|\alpha - \beta|_v \ge (3h(\alpha))^{-\kappa_1} h(\beta)^{-\kappa_2}$ (where $|\cdot|_v$ is the absolute value at a place $v$ of a number field containing $\alpha$ and $\beta$, and $\kappa_1, \kappa_2 > 0$ are explicit exponents depending on the degree $r$ and the method's parameters) holds for all but finitely many pairs $(\alpha, \beta)$, or the heights of the remaining exceptional $\beta$ are confined to an explicit interval; the method becomes fully effective once one exhibits a single anchor $\beta$ that approximates $\alpha$ so well that it lands among the exceptions, while $h(\beta)$ is large enough relative to $h(\alpha)$. The paper's Section V constructs anchors for special families - for example, for the real root of $x^r - m x^{r-1} + 1 = 0$ the rational $1/m$ approximates $\alpha^{-1}$ to order roughly $r$ - giving completely explicit effective exponents (e.g. 39.2574 for degree $r \ge 40$ in Example 3, and $(1 - 10^{-4})r$ for $m \ge 2561$ and effectively large $r$ in Example 5).

Roth's theorem itself, with exponent $2 + \varepsilon$ for all algebraic $\alpha$, uses several simultaneous approximations $p_1/q_1, \dots, p_m/q_m$ and a multivariate zero estimate now called Roth's lemma, which requires the heights of the approximations to grow rapidly (geometrically) from one to the next. This height-growth condition is precisely where effectivity dies: the gap between consecutive heights leaves the argument unable to enumerate the exceptional approximants, and the final constant absorbs an unknown finite set. Viola (1985) records Bombieri's observation that Dyson's method, unlike Siegel's and Roth's, is essentially free from considerations of heights - the basis of the hope that the Thue-Siegel-Dyson route could be made effective where Roth's route apparently cannot.

The anchor problem is intertwined with the continued-fraction expansion of $\alpha$: an anchor for a general algebraic number would require partial control of the continued fraction, where even the unboundedness of partial quotients for algebraic numbers of degree $\ge 3$ is a famous open problem dating from the 1960s. This is why all known anchor constructions work only for families whose defining equations force an exceptionally good rational approximation algebraically.

The 1982 paper's concluding discussion isolates four questions, reproduced in the problem statement below: the first, third and fourth concern effectivity (the auxiliary height bound, the height-growth condition, and the anchor construction), while the second - the several-variable Dyson lemma - was expected to let several independent approximations enter the argument as in Roth's proof.

Problem Statement

Let $\alpha$ be an irrational algebraic number of degree $r \ge 3$, and let $h(\cdot)$ denote the logarithmic Weil height (for a rational $p/q$ in lowest terms, $h(p/q) = \log \max(|p|, q)$; the multiplicative height is $H = \exp h$, so $H(p/q) = \max(|p|, q)$). The problem, raised by the concluding discussion of Bombieri's 1982 paper On the Thue-Siegel-Dyson theorem, is to make the Thue-Siegel-Dyson method effective - that is, to replace the non-computable constants in irrationality-measure bounds by constants computable from the data - ideally achieving an effective version of Roth's theorem: for every irrational algebraic $\alpha$ and every $\varepsilon > 0$, an algorithmically computable $c(\alpha, \varepsilon) > 0$ such that $|\alpha - p/q| > c(\alpha, \varepsilon)\, H(p/q)^{-2-\varepsilon}$ for all rationals $p/q$. The source discussion isolates four questions:

(Q1) A sharpening of Lemma 6 of that paper - the explicit height bound for the auxiliary polynomial, linear in the bidegrees $d_1, d_2$ weighted by height terms of $\alpha_1$ and $\alpha_2$, with an explicit coefficient depending on the degree $r$ and on auxiliary parameters of the construction - could lead to the optimal exponent 2 in every case: an effective version of Roth's theorem.

(Q2) An extension of Dyson's lemma (the two-variable zero estimate) to more than two variables would allow several independent rational approximations $p_1/q_1, \dots, p_m/q_m$ to enter the argument, as in Roth's proof, and might circumvent the large-height requirements that now limit the method.

(Q3) A formulation of Roth's lemma - the multivariate zero estimate at the heart of Roth's proof - that does not require the approximations to have rapidly increasing heights, needed to turn Roth's theorem into an effective result. The source paper offers no contribution to this problem.

(Q4) The construction of explicit pairs $(\alpha, \beta)$ with $\beta$ an extremely good approximation to $\alpha$ and with $h(\beta)$ sufficiently large relative to $h(\alpha)$ - so that the hypotheses of the effective method (the alternative of Bombieri's Theorem 4) are satisfied: $\beta$ must approximate $\alpha$ so well that it violates the power-law bound $|\alpha - \beta|_v \ge (3h(\alpha))^{-\kappa_1} h(\beta)^{-\kappa_2}$ - where $|\cdot|_v$ is the absolute value at a place $v$ of a number field containing $\alpha$ and $\beta$, and $\kappa_1, \kappa_2 > 0$ are the explicit exponents of Theorem 4, functions of the degree $r$ and the method's parameters - while $h(\beta)$ is large. The Section V examples of the source paper rely on an algebraic deformation that forces an initial approximation (roots of $x^r - m x^{r-1} + 1 = 0$); for a general algebraic number no such construction is known.

Since (Q2) is resolved as a lemma, the surviving open core is, in the general case of all algebraic numbers (or at least of all algebraic numbers of degree above an explicit threshold):

(i) an effective irrationality measure for all algebraic numbers with the optimal exponent 2 (effective Roth's theorem), or at minimum an effective Thue-Siegel-Dyson bound with exponent $(1+o(1))\sqrt{2r}$ (as $r \to \infty$) whose constants are computable from the degree and height of $\alpha$, valid for all algebraic numbers (not only special fields);

(ii) a version of Roth's lemma free of the requirement that the approximations' heights grow rapidly from one to the next, with effectively computable constants;

(iii) for a general algebraic number $\alpha$, an explicit construction - or a proof of existence with computable data - of an anchor pair $(\alpha, \beta)$ satisfying the inequalities of (Q4).

References

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