Roth's Theorem Research Articles

Diophantine Approximation and Continued Fraction Expansions of Algebraic Power Series in Positive Characteristic

In a recent paper M. Buck and D. Robbins have given the continued fraction expansion of an algebraic power series when the base field is F3. We study its rational approximation property in relation with Roth's theorem, and we show that this element has an analog for each power of an odd prime number. At last we give the explicit continued fraction expansion of another classical example.

The extension of Roth's theorem for matrix equations over a ring

This paper extends Roth's similarity theorem as follows: Let R be a ring with identity, B( λ) = Σ k i = 0 B i λ i ϵ R m × q [ λ]. If either R is a division ring and A ϵ R n × n is algebraic, or R is finitely generated as a module over its center, then the matrix equation Σ k i = 0 A i XB i = C over R has a solution if and only if ▪

Autour du th�or�me de Roth

On Roth's theorem. The celebrated theorem of Roth, together with its generalizations given by Mahler and Ridout, gives a lower bound for the degree of approximation of one or more algebraic numbers with respect to a fixed set of valuations by elements of a fixed number field. An analogous result holds for function fields in characteristic zero. In this paper we do the following: (1) generalize Roth's theorem to the case of fields with a product formula in characteristic zero, removing any technical hypothesis from a previous result of Lang: (2) give a unified proof of Roth's theorem in the number field and function field cases; (3) provide a quantitative version of the general Roth's theorem, extending, even in the number field case, previous results of Bombieri and Van Der Poorten.

An Effective Roth's Theorem for Function Fields

An Effective Roth's Theorem for Function Fields

The matrix equation AXB− GXD = E over the quaternion field

Using the method of complex representation, this paper obtains necessary and sufficient conditions for the existence of a solution or a unique solution to the matrix equation AXB − GXD = E over the quaternion field, extends W. E. Roth's theorem and A. Jameson's theorem to the quaternion field, and gives the representation of the solution.

The solvability of linear matrix equations over a central simple algebra

Let R be a finite dimensional central simple algebra over a field F. By Using the method of representation, this paper discusses the linear matrix equations over R, obtains several necessary and sufficient conditions for existence of a solution or a unique solution, gives several explicit formulas of the unique solution for the equation AX − XB = C or X − AXB = C over R, and extends a Roth's theorem to the matrix equation over R.

Roth's theorem with moving targets

Roth's theorem with moving targets

The number of exceptional approximations in Roth's theorem

AbstractRoth's Theorem says that given ρ < 2 and an algebraic number α, all but finitely many rational numbers x/y satisfy |α - (x/y)|< |y|-ρ. We give upper bounds for the number of these exceptional rationals when 3 ≤ ρ ≤ d, where d is the degree of α. Our result suplements bounds given by Bombieri and Van der Poorten when 2 > ρ ≤ 3; naturally the bounds become smaller as ρ increases.

Uniform approximation by solutions of elliptic equations with continuous extension to the boundary

Let F be a relatively closed subset of an open set G of the plane. Alice Roth in [10] proved that if a function f is a uniform limit on F of holomorphic or meromorphic functions on G, it is possible to select the approximating functions m in such a way that the difference function f m can be extended continuously into the boundary of F. In this paper we consider the more general situation arising when one replaces ∂[bbar] by an elliptic operator p(D) with constant coefficients. We prove that the natural analog of Roth's theorem for the operator P(D) holds, at least for bounded open sets G. In those cases where one can find an inversion of the domain, preserving the uniform approximation by solutions of the operator, unbounded open sets are allowed too. This hannens when P(D)=δ and n=2. In this case we improve the main result of Goldstein and O w in [7], removing most oi the unnecessary conditions assumed in their work.

Roth's theorems for matrix equations with symmetry constraints

This note deals with the consistency of complex matrix equations AX − YB = C and AX − XB = C under the constraints Y = X ∗ and X = X ∗ .

Irrationality Measures of Carlitz Zeta Values in Characteristic p

We calculate exact measures of irrationality of certain Carlitz zeta values. By using a conjecture about the "Roth Theorem" in characteristic p, we show that these results lead to transcendence results.

A note on Roth's Theorem

We give a quantitative version of Roth's Theorem over an arbitrary number field, similar to that given by Bombieri and van der Poorten.

Some quantitative results related to Roth's theorem: Corrigenda

Theorem 1 of the paper [1] is stated in an unnecessarily weak form. What is actually proved is much stronger.

Herbrand-Analysen zweier Beweise des Satzes von Roth: Polynomiale Anzahlschranken

AbstractA previously unexplored method, combining logical and mathematical elements, is shown to yield substantial numerical improvements in the area of Diophantine approximations. Kreisel illustrated the method abstractly by noting that effective bounds on the number of elements are ensured if Herbrand terms from ineffective proofs ofΣ2-finiteness theorems satisfy certain simple growth conditions. Here several efficient growth conditions for the same purpose are presented that are actually satisfied in practice, in particular, by the proofs of Roth's theorem due to Roth himself and to Esnault and Viehweg. The analysis of the former yields an exponential bound of order exp(70ε−2d2) in place of exp(285ε−2d2) given by Davenport and Roth in 1955, whereαis (real) algebraic of degreed≥ 2 and ∣α−pq−1∣ <q−2−ε. (Thus the new bound is less than the fourth root of the old one.) The new bounds extracted from the other proof arepolynomial of low degree(inε−1and logd). Corollaries: Apart from a new bound for the number of solutions of the corresponding Diophantine equations and inequalities (among them Thue's inequality), log logqν, <Cα, εν5/6+ε, whereqνare the denominators of the convergents to the continued fraction ofα.

A nonlinear version of Roth's theorem for sets of positive density in the real line

It is proved that given e>0, there is δ(e)>0 such that ifS is a measurable set of [0,N], |S|>eN, then there is a triplex, x+h, x+h2 inS withh satisfyingh>δ(e)N1/2. The argument is related to [B] and uses the behavior of certain non-linear convolution-type operators. The method can be adapted in a variety of situations. For instance, it can be used to prove the analogue of the previous statement with the square replaced by another power,h3,h4 etc.

Some quantitative results related to Roth's Theorem

AbstractWe employ the Dyson's Lemma of Esnault and Viehweg to obtain a new and sharp formulation of Roth's Theorem on the approximation of algebraic numbers by algebraic numbers and apply our arguments to yield a refinement of the Davenport-Roth result on the number of exceptions to Roth's inequality and a sharpening of the Cugiani-Mahler theorem. We improve on the order of magnitude of the results rather than just on the constants involved.

The matrix equation X − AXB = C and an analogue of Roth's theorem

We give a necessary and sufficient condition for the consistency of the matrix equation X − AXB = C.

Roth's theorems for sets of matrices

It is shown that Roth's theorems on the equivalence and similarity of block diagonal matrices hold for finite sets of matrices over a commutative ring.

Lines imply spaces in density Ramsey theory

Some results of geometric Ramsey theory assert that if F is a finite field (respectively, set) and n is sufficiently large, then in any coloring of the points of F n there is a monochromatic k-dimensional affine (respectively, combinatorial) subspace (see [9]). We prove that the density version of this result for lines (i.e., k = 1) implies the density version for arbitrary k. By using results in [3, 6] we obtain various consequences: a “group-theoretic” version of Roth's Theorem, a proof of the density assertion for arbitrary k in the finite field case when ∥ F∥ = 3, and a proof of the density assertion for arbitrary k in the combinatorial case when ∥ F∥ = 2.

Rational approximations to solutions of linear differential equations

Rational approximations of Padé and Padé type to solutions of differential equations are considered. One of the main results is a theorem stating that a simultaneous approximation to arbitrary solutions of linear differential equations over C(x) cannot be "better" than trivial ones implied by the Dirichlet box principle. This constitutes, in particular, the solution in the linear case of Kolchin's problem that the "Roth's theorem" holds for arbitrary solutions of algebraic differential equations. Complete effective proofs for several valuations are presented based on the Wronskian methods and graded subrings of Picard-Vessiot extensions.