Roth's Theorem Research Articles

Roth's theorem for ruled surfaces

This paper addresses conjectures of E. Bombieri and P. Vojta in the special case of ruled surfaces not birational to [inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="01i" /]. Apart from this implicit restriction to [inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="02i" /] bundles S over an elliptic curve, the ultimate question of the arithmetic of pairs ( S, D ) for a divisor D requires further restrictions on D which turn the proposed conjectures into the study of Roth's theorem on approximation of algebraic numbers α, but for α now parametrized by an elliptic curve. With these restrictions, best possible answers are obtained. The same study may also be carried out for holomorphic maps, and this is done simultaneously.

On a general Thue's equation

We analyze the integral points on varieties defined by one equation of the form f 1 · · · f r = g , where the f i , g are polynomials in n variables with algebraic coefficients, and g has "small" degree; we shall use a method that we recently introduced in the context of Siegel's Theorem for integral points on curves. Classical, very particular, instances of our equations arise, e.g., in a well-known corollary of Roth's Theorem (the case n = 2, f i linear forms, deg g < r -2) and with the norm-form equations , treated by W. M. Schmidt. Here we shall prove (Thm. 1) that the integral points are not Zariski-dense, provided Σdeg f i > n · max (deg f i ) + deg g and provided the f i , g , satisfy certain (mild) assumptions which are "generically" verified. Our conclusions also cover certain complete-intersection subvarieties of our hypersurface (Thm. 2). Finally, we shall prove (Thm. 3) an analogue of the Schmidt's Subspace Theorem for arbitrary polynomials in place of linear forms.

Distances between the conjugates of an algebraic number

Given a fixed number field K, we give non-trivial lower bounds for the distance between the conjugates of any number a from K in terms of the Mahler measure M(a) of a. In the case that K is a cubic field, our bound is best possible in terms of M(a). Our proof is based on Roth's theorem.

On Roth's theorem concerning a cube and three cubes of primes

On Roth's theorem concerning a cube and three cubes of primes

On Roth's theorem concerning a cube and three cubes of primes

In this paper, we prove that with at most O(N1271/1296+ε) exceptions, all positive integers up to N are the sum of a cube and three cubes of primes. This improves an earlier result O(N169/170) of the first author and the classical result O(NL‐A) of Roth.

Failure of the Roth theorem for solvable groups of exponential growth

We show that for any finitely generated solvable group of exponential growth one can find a measure-preserving action for which the multiple recurrence theorem fails and a measure-preserving action for which the ergodic Roth theorem fails. This is in contrast with the positive results established by Leibman (Geom. Funct. Anal.8 (1998), 853–931) and Bergelson and Leibman (Invent. Math.147 (2002), 429–470) for nilpotent group actions.

On the rational approximations to the powers of an algebraic number: Solution of two problems of Mahler and Mendès France

About fifty years ago Mahler [Ma] proved that i f ~ > 1 is rational but not an integer and i f 0 l all of whose conjugates different from cr have absolute value less than 1 (note that rational integers larger than 1 are Pisot numbers according to our definition). Mahler also added that It would be of some interest to know which algebraic numbers have the same property as [the rationals in the theorem]. Now, it seems that even replacing Ridout 's theorem with the modern versions of Roth 's theorem, valid for several valuations and approximations in any given number field, the method of Mahler does not lead to a complete solution to his question. One of the objects of the present paper is to answer Mahler 's question completely; our methods will involve a suitable version of the Schmidt subspace theorem, which may be considered as a multi-dimensional extension of the results mentioned by Roth, Mahler and Ridout. We state at once our first theorem, where as usual we denote by IIx]] the distance of the complex number x from the nearest integer in Z, i.e.

An extension of a result of Sierpiński

As an application of Roth's theorem concerning the rational approximation of algebraic numbers, two sufficiency conditions are derived for an alternating series of rational terms to converge to a transcendental number. The first of these conditions represents an extension of an earlier condition of Sierpiński for the convergence of alternating series to irrational values.

The ABC Conjecture Implies Vojta's Height Inequality for Curves

Following Elkies (Internat. Math. Res. Notices7 (1991) 99–109) and Bombieri (Roth's theorem and the abc-conjecture, preprint, ETH Zürich, 1994), we show that the ABC conjecture implies the one-dimensional case of Vojta's height inequality. The main geometric tool is the construction of a Belyǐ function. We take care to make explicit the effectivity of the result: we show that an effective version of the ABC conjecture would imply an effective version of Roth's theorem, as well as giving an (in principle) explicit bound on the height of rational points on an algebraic curve of genus at least two.

The ABC Conjecture Implies Vojta's Height Inequality for Curves

Following Elkies (Internat. Math. Res. Notices7 (1991) 99–109) and Bombieri (Roth's theorem and the abc-conjecture, preprint, ETH Zürich, 1994), we show that the ABC conjecture implies the one-dimensional case of Vojta's height inequality. The main geometric tool is the construction of a Belyǐ function. We take care to make explicit the effectivity of the result: we show that an effective version of the ABC conjecture would imply an effective version of Roth's theorem, as well as giving an (in principle) explicit bound on the height of rational points on an algebraic curve of genus at least two.

A nilpotent Roth theorem

Let T and S be invertible measure preserving transformations of a probability measure space (X, ℬ, μ). We prove that if the group generated by T and S is nilpotent, then $\lim_{N\ras\infty}\frac{1}{N}\sum_{n=1}^{N}u(T^{n}x)v(S^{n}x)$ exists in L 2-norm for any u, v∈L ∞(X, ℬ, μ). We also show that for A∈ℬ with μ(A)>0 one has $\lim_{N\ras\infty}\frac{1}{N}\sum_{n=1}^{N} \mu(A\cap T^{-n}A\cap S^{-n}A)>0$ . By the way of contrast, we bring examples showing that if measure preserving transformations T, S generate a solvable group, then (i) the above limits do not have to exist; (ii) the double recurrence property fails, that is, for some A∈ℬ, μ(A)>0, one may have μ(A∩T -n A∩S - n A)=0 for all n∈ℕ. Finally, we show that when T and S generate a nilpotent group of class ≤c, $\lim_{N\ras\infty}\frac{1}{N}\sum_{n=1}^{N}u(T^{n}x)v(S^{n}x) =\int ud\mu\int vd\mu$ in L 2(X) for all u, v∈L ∞(X) if and only if T×S is ergodic on X×X and the group generated by T -1 S, T -2 S 2,..., T -c S c acts ergodically on X.

A subspace theorem approach to integral points on curves

We present a proof of Siegel's theorem on integral points on affine curves, through the Schmidt subspace theorem, rather than Roth's theorem. This approach allows one to work only on curves, avoiding the embedding into Jacobians and the subsequent use of tools from the arithmetic of Abelian varieties. To cite this article: P. Corvaja, U. Zannier, C. R. Acad. Sci. Paris, Ser. I 334 (2002) 267–271.

A Theorem on Transcendence of Infinite Series II

As an application of Roth's theorem concerning the rational approximation of algebraic numbers, a sufficiency condition is derived for a series of positive rational terms to converge to a transcendental number. This condition is then used to obtain similar sufficiency conditions that exist within the literature. In addition, as a consequence of the sufficiency condition obtained we exhibit some transcendental valued series involving generalised Fibonacci and Lucas numbers.

The exceptional set in Roth's theorem concerning a cube and three cubes of primes

The exceptional set in Roth's theorem concerning a cube and three cubes of primes

A Weak Effective Roth's Theorem over Function Fields

A Weak Effective Roth's Theorem over Function Fields

A constructive proof by elementary transformations of Roth's removal theorems in the theory of matrix equations

W. E. Roth gave necessary and sufficient conditions for the existence of solution(s) of certain types of linear matrix equations. Proofs were based on invariant factors and were long and complicated. Other shorter but non-constructive proofs have since been provided by later authors. We present here very brief constructive proofs based on the simplest of mathematical techniques, namely row- and column-reduction of a matrix.

Diophantine approximation and deformation

It is well-known that while the analogue of Liouville's theorem on diophan- tine approximation holds in finite characteristic, the analogue of Roth's theorem fails quite badly. We associate certain curves over function fields to given algebraic power series and show that bounds on the rank of Kodaira-Spencer map of this curves imply bounds on the diophantine approximation exponents of the power series, with more 'generic' curves (in the de- formation sense) giving lower exponents. If we transport Vojta's conjecture on height inequality to finite characteristic by modifying it by adding suitable deformation theoretic condition, then we see that the numbers giving rise to general curves approach Roth's bound. We also prove a hierarchy of exponent bounds for approximation by algebraic quantities of bounded degree. R ´ ESUM´ E.— APPROXIMATION DIOPHANTIENNE ET D´ EFORMATION. — Alors que l'analogue

A straight forward proof of Roth's lemma in matrix equations

In 1952 W. E. Roth proved two theorems on matrix equations using a lemma far more profound than the theorems. Others have produced simpler proofs of the theorems but not of Roth's lemma. We present a method that proves Roth's lemma easily, and is a powerful tool in its own right, enabling us to extend Roth's theorems by giving insight into their underlying structure.

Roth's removal theorem for lambda matrices

Roth's removal theorem is a fundamental result in linear algebra which gives necessary and sufficient conditions for Sylvester's matrix equation to have a solution. Here we show that Roth's theorem is true for lambda matrices and integer matrices thus greatly extending the area of validity of this fundamental result.

A Roth theorem for amenable groups

We prove the following mean ergodic theorem: for any two commuting measure preserving actions { T g } and { S g } of a countable amenable group G on a probability space ( X, A , μ), lim n →∞ 1/|Φ n | Σ g ∈Φ n φ( T g x )ψ( S g T g x ) exist in L 1 ( X, A , μ) for any φ, ψ, ∈ L 2 ( X, A , μ), where {Φ n } is any left Følner sequence for G . This generalizes Furstenberg's ergodic Roth theorem, which corresponds to the case G = Z , T g = S g , as well as a more general result of Conze and Lesigne (which corresponds to the case G = Z with no restrictions on T g and S g ). The limit is identified, and two combinatorial corollaries are obtained. The first of these states that in any subset E ⊂ G × G which is of positive upper density (with regard to any left Følner sequence in G × G ), we may find triangular configurations of the form {( a, b ), ( ga, b ), ( ga, gb )}. This result has as corollaries Roth's theorem on arithmetic progressions of length three and a theorem of Brown and Buhler guaranteeing solutions to the equation x + y = 2 z in any sufficiently big subset of an abelian group of odd order. The second corollary states that if G × G × G is partitioned into finitely many cells, one of these cells contains configurations of the form {( a, b, c ), ( ga, b, c ,), ( ga, gb, c ), ( ga, gb, gc )}.