Introduction to Diophantine Approximation. Part II

Proceedings of the National Academy of Sciences, 1984

This paper is devoted to the study of the arithmetic properties of values of G -functions introduced by Siegel [Siegel, C. L. (1929) Abh. Preuss. Akad. Wiss. Phys.-Math. Kl. 1]. One of the main results is a theorem on the linear independence of values of G -functions at rational points close to the origin. In this theorem, no conditions are imposed on the p -adic convergence of a G -function at a generic point. The theorem finally realizes Siegel's program on G -function values outlined in his paper.

Journal of Pure and Applied Algebra, 1997

We introduce a subexponential algorithm for geometric solving of multivariate polynomial equation systems whose bit complexity depends mainly on intrinsic geometric invariants of the solution set. ¿From this algorithm, we derive a new procedure for the decision of consistency of polynomial equation systems whose bit complexity is subexponential, too. As a byproduct, we analyze the division of a polynomial modulo a reduced complete intersection ideal and from this, we obtain an intrinsic lower bound for the logarithmic height of diophantine approximations to a given solution of a zero-dimensional polynomial equation system. This result represents a multivariate version of Liouville's classical theorem on approximation of algebraic numbers by rationals. A special feature of our procedures is their polynomial character with respect to the mentioned geometric invariants when instead of bit operations only arithmetic operations are counted at unit cost. Technically our paper relies on the use of straight-line programs as a data structure for the encoding of polynomials, on a new symbolic application of Newton's algorithm to the Implicit Function Theorem and on a special, basis independent trace formula for affine Gorenstein algebras.

International Journal of Mathematics and Mathematical Sciences, 2005

We prove the existence of a dense subset ∆ of [0,4] such that for all α ∈ ∆ there exists a subgroup X α of infinite rank of Z[z] such that X α is a discrete subgroup of C[0,β] for all β ≥ α but it is not a discrete subgroup of C[0,β] for any β ∈ (0,α).

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Journal of Number Theory, 1986

1996

We introduce a subexponential algorithm for geometric solving of multivariate polynomial equation systems whose bit complexity depends mainly on intrinsic geometric invariants of the solution set. From this algorithm, we derive a new procedure for the decision of consistency of polynomial equation systems whose bit complexity is subexponential, too. As a byproduct, we analyze the division of a polynomial modulo

Journal of Modern Dynamics, 2007

Let Q be a nondegenerate indefinite quadratic form on R n , n ≥ 3, which is not a scalar multiple of a rational quadratic form, and let C Q = {v ∈ R n | Q(v) = 0}. We show that given v 1 ∈ C Q , for almost all v ∈ C Q \ Rv 1 the following holds: for any a ∈ R, any affine plane P parallel to the plane of v 1 and v, and > 0 there exist primitive integral n-tuples x within distance of P for which |Q(x) − a| <. An analogous result is also proved for almost all lines on C Q .

Ann. Math. Inform, 2008

In this paper we prove several results on connection between continued fractions and rational approximations of the form |α−a/b| < k/b 2 , for a positive integer k.

Glasgow Mathematical Journal, 1996

In [5] Professor Hooley announced without proof the following result which is a variant of well-known work by Heilbronn [4]and Danicic [3] (see [1]).Let k≥2 be an integer, b a fixed non-zero integer, and a an irrational real number. Then, for any ɛ&gt; 0, there are infinitely many solutions to the inequalityHere

Pacific Journal of Mathematics, 1982

Proceedings - Mathematical Sciences, 2009

Given a sequence (x n) ∞ n=1 of real numbers in the interval [0, 1) and a sequence (δ n) ∞ n=1 of positive numbers tending to zero, we consider the size of the set of numbers in [0, 1] which can be 'well approximated' by terms of the first sequence, namely, those y ∈ [0, 1] for which the inequality |y − x n | < δ n holds for infinitely many positive integers n. We show that the set of 'well approximable' points by a sequence (x n) ∞ n=1 , which is dense in [0, 1], is 'quite large' no matter how fast the sequence (δ n) ∞ n=1 converges to zero. On the other hand, for any sequence of positive numbers (δ n) ∞ n=1 tending to zero, there is a well distributed sequence (x n) ∞ n=1 in the interval [0, 1] such that the set of 'well approximable' points y is 'quite small'.

In this note, we show that if α is a real number such that there exist a constant c and a sequence of non-zero integers (rn)n≥0 with limn→∞ |rn| = 1 for which �� �α rn+1 rn � � � &lt; c |rn|2 holds for all n � 0, then either α 2 Z\{0, ±1} or α is a quadratic unit. Our result complements results obtained by P. Kiss who established the converse in Period. Math. Hungar. 11 (1980), 281-187.

Journal of the London Mathematical Society, 1989

Many variations of the theorems of Dirichlet and Kronecker in Diophantine approximation [3, Chapters 11.3 and 23] have been established, by restricting the class of integers allowed in the problem. For example, see [1], where one is restricted to the values taken by a polynomial with integer coefficients. Another example is afforded by the following theorem. Suppose that a is irrational and let \\y\\ denote the smallest distance of y from an integer. Then, for any real number ft, there are infinitely many primes p such that \\ap-p\\<p-3 ' 10. (1.1) This result was established in [5], improving earlier results of Vinogradov [11, Chapter 11] and Vaughan [9]. In this paper we shall consider extending this result to simultaneous approximation. In [4, Chapter 3] the author considered this problem for sets of not very well approximable numbers (see [7], or below for the definition) and this paper develops the author's previously unpublished work. Balog and Friedlander have recently done some work on this question [2], but our results supersede theirs in all cases. We wish to prove that max is ' small' for infinitely many primes p. A necessary restriction on a 15 ..., OL S is given by the following definition due to Balog and Friedlander. A set of real numbers {a l5 ..., a s } is called compatible if s s £ «, a, e Q implies that ^^a^e Z whenever n lt ...n s are integers. Balog and Friedlander then proved the following. Let {a 15 ..., a s } be a compatible set of real algebraic numbers, lying in a field of degree d. Then, for any A < (3ds + d-s-\)~1, there are infinitely many solutions in primes p to max || pa, || <p~A. We improve this result as follows.

ACTA ARITHMETICA-WARSZAWA-, 2006

Journal of Number Theory, 1974

We sharpen a technique of Gelfond to show that, in a sense, the only possible gap-free sequences of "good" Diophantine approximations to a fixed Al E C are trivial ones. For example, suppose that a > 1 and that (S,)~=, and (un)FE1 are two positive, strictly increasing unbounded sequences satisfying IS