Roth's Theorem Research Articles

Roth's removal rule revisited

The theory of companion matrices is used to give explicit representations for the matrices needed in Roth's removal rule. These are then used to give simple proofs for the cyclic decomposition theorem, as well as for Roth's similarity theorem for matrices over a field.

Roth's theorems and decomposition of modules

It is shown that there is a connection between Roth's theorems on similarity and equivalence of block-triangular matrices and decomposition of modules. The module property is that if M≅ N⊕M N , then N is a summand of M . This holds for any commutative ring if M is finitely presented. New proofs of Roth's theorems are given for commutative rings. Some results are established in the noncommutative case.

On matrix equivalence and matrix equations

Roth's theorem on the solvability of matrix equations of the form AX− YB= C is proved for matrices with coefficients from a division ring or a ring which is module-finite over its center.

Roth's theorems over commutative rings

In 1952, W.E. Roth showed that matrix equations of the forms AX− YB = C and AX− XB = C over fields can be solved if and only if certain block matrices built from A, B, and C are equivalent or similar. We show here that these criteria remain valid over arbitrary commutative rings. To do this, we use standard commutative algebra methods to reduce to the case of Artinian rings, where a simple argument with

Periodic distributions and non-Liouville numbers

In an earlier paper (Translation-invariant linear forms and a formula for the Dirac measure, J. Functional Analysis 8 (1971), 173–188) the author proved that the ratio α β of two real numbers is a non-Liouville number if and only if there exist two Schwartz distributions A and B with compact supports on the real line such that δ′ = A − τ α A + B − τ β B. The present paper presents (in Section 3) a completely new and more elementary proof of this result (stated fully as Theorems 1 and 2 at the end of Section 1) based on some fundamental properties of the mapping H 0 = − Δ and H = H 0 + T, where T which are established in Section 2. Further connections with Diophantine approximation (badly approximable numbers and Roth's Theorem) are presented in Section 4 where it is proved that the orders of the distributions A and B are always ⩾2 (Theorem 3) and almost always =3 (Theorem 4). Section 5 contains some partial results (Theorem 5 and Corollary 2 of Theorem 6) on the analogous question (as yet unsettled) for the Banach space C(T) of all continuous functions on the circle group T, and connects this problem with Sidon sets of integers.

On norm form equations

In his paper [ Ann. of Math. 96 (1972)] Schmidt applied his results on the n-dimensional Roth theorem to study the n-dimensional Thue equation f( x 1,…, x n ) = c, and he proved, when f is of norm type, a necessary and sufficient criterion for the Thue equation to have only finitely many solutions. Here we shall study equations f( x 1,…, x n ) = p 1 z 1 … p t z t and prove the p-adic analog of Schmidt's theorems.

Roth’s equivalence problem in unit regular rings

It is shown that for matrices over a unit regular ring [ A a m p ; C 0 a m p ; D ] ∼ [ A a m p ; 0 0 a m p ; D ] \left [ {\begin {array}{*{20}{c}} A & C \\ 0 & D \\ \end {array} } \right ]\sim \left [ {\begin {array}{*{20}{c}} A & 0 \\ 0 & D \\ \end {array} } \right ] if and only if there exist solutions X X and Y Y to A X − Y D = C AX - YD = C , thus providing a partial generalization to Roth’s theorem.

Fractional parts of powers of rationals

Mahler (5) proved in 1957 that for any rational a/q, where a, q are relatively prime integers with a > q ≥ 2, and any ε > 0, there exist only finitely many positive integers n such that ∥(a/q)n∥ < e−εn; here ∥x∥ denotes the distance of x from the nearest integer taken positively. In particular there exist only finitely many n such thatand, as Mahler observed, this implies that the number g(k) occurring in Waring's problem is given byfor all but a finite number of values of k. It would plainly be of interest to establish a bound for the exceptional k and this would follow from an upper estimate for the integers n for which (1) holds. But Mahler's work was based on Ridout's generalization of Roth's theorem and, as is well known, the latter result is ineffective.

The geometry of the riccati equation

A theory for the determination of all symmetric solutions of the Riccati equation is presented, based on a generalized version of the Bass–Roth theorem. Further disconjugacy results for a general variational problem are given.

Linear forms with algebraic coefficients. I

In a recent paper I generalized Roth's well-known theorem on rational approximation to an algebraic number α in two dual directions. Namely, I proved a theorem on simultaneous rational approximation to m algebraic numbers α 1,…, α m , and a theorem on linear forms L( x) = α 1 x 1 + … + α m x m + x m+1 . In the present paper I shall carry the argument further and I shall prove similar results for systems of linear forms.

An application of the Thue–Siegel–Roth theorem to elliptic functions

Let α1, …, αn be n ≥ 2 algebraic numbers such that log α1,…, log αn and 2πi are linearly independent over the field of rational numbers Q. It is well known (see (6), Ch. 1) that the Thue–Siegel–Roth theorem implies that, for each positive number δ, there are only finitely many integers b1,…, bn satisfyingwhere H denotes the maximum of the absolute values of b1, …, bn. However, such an argument cannot provide an explicit upper bound for the solutions of (1), because of the non-effective nature of the theorem of Thue–Siegel–Roth. An effective proof that (1) has only a finite number of solutions was given by Gelfond (6) in the case n = 2, and by Baker(1) for arbitrary n. The work of both these authors is based on arguments from the theory of transcendental numbers. Baker's effective proof of (1) has important applications to other problems in number theory; in particular, it provides an algorithm for solving a wide class of diophantine equations in two variables (2).

Linear forms in the logarithms of algebraic numbers

In 1934 Gelfond [2] and Schneider [6] proved, independently, that the logarithm of an algebraic number to an algebraic base, other than 0 or 1, is either rational or transcendental and thereby solved the famous seventh problem of Hilbert. Among the many subsequent developments (cf. [4, 7, 8]), Gelfond [3] obtained, by means of a refinement of the method of proof, a positive lower bound for the absolute value of β1 log α1+β2 log α2, where β1, β2 denote algebraic numbers, not both 0, and α1,α1, denote algebraic numbers not 0 or 1, with log α1/log α2 irrational. Of particular interest is the special case in which β1,β2 denote integers. In this case it is easy to obtain a trivial positive lower bound (cf. [1; Lemma 2]), and the existence of a non-trivial bound follows from the Thue–Siegel–Roth theorem (see [4; Ch. I]). But Gelfond's result improves substantially on the former, and, unlike the latter, it is derived by an effective method of proof. Gelfond [4; p. 177] remarked that an analogous theorem for linear forms in arbitrarily many logarithms of algebraic numbers would be of great value for the solution of some apparently very difficult problems of number theory. It is the object of this paper to establish such a result.

On the approximation of algebraic numbers by algebraic integers

In his Topics in Number Theory, vol. 2, chapter 2 (Reading, Mass., 1956) W. J. LeVeque proved an important generalisation of Roth's theorem (K. F. Roth, Mathematika 2,1955, 1—20).Let ξ be a fixed algebraic number, σ a positive constant, and K an algebraic number field of degree n. For κ∈K denote by κ(1), …, κ(n) the conjugates of κ relative to K, by h(κ) the smallest positive integer such that the polynomial has rational integral coefficients, and by q(κ) the quantity

Continued fractions of transcendental numbers

It is well known that if, in a continued fractionthere is a subsequence of the an which increases very rapidly, then ξ is a transcendental number. A result of this kind follows from Liouville's Theorem on rational approximations to algebraic numbers, but the most precise result so far established is that which was deduced from Roth's Theorem by Davenport and Roth [1]. They proved (Theorem 3) that if ξ is algebraic, thenwhere qn is the denominator of the nth convergent to (1). Thus ifξ is transcendental.