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Some Applications of Diophantine Approximation
 Math. Inst. Leiden, Report MI
, 2000
"... The paper gives a survey of some results on diophantine approximation (Sections 1 and 2) and their applications (Sections 3,4 and 5). Section 1 contains an introduction to the theory of linear forms in logarithms of algebraic numbers and Section 2 some results following from the Subspace Theorem ..."
Abstract

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The paper gives a survey of some results on diophantine approximation (Sections 1 and 2) and their applications (Sections 3,4 and 5). Section 1 contains an introduction to the theory of linear forms in logarithms of algebraic numbers and Section 2 some results following from the Subspace Theorem. In Section 3 we consider the local behaviour of sequences of numbers composed of small primes and of sums of two such numbers and in Section 4 the transcendence of innite sums of values of a rational function and related sums.
Congruence properties of the Ωfunction on sumsets
 Illinois J. Math
, 1999
"... constants. Z, N and N0 denote the set of integers, positive integers and nonnegative integers respectively. The cardinality of a set S is denoted by S. ⌊x ⌋ and {x} denote the integer part and the fractional part of x and ‖x ‖ denotes the distance from x to the nearest integer: ‖x ‖ = min({x}, 1 ..."
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Cited by 4 (1 self)
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constants. Z, N and N0 denote the set of integers, positive integers and nonnegative integers respectively. The cardinality of a set S is denoted by S. ⌊x ⌋ and {x} denote the integer part and the fractional part of x and ‖x ‖ denotes the distance from x to the nearest integer: ‖x ‖ = min({x}, 1 − {x}). We write e 2πiα = e(α). If f(n) = O(g(n)), then we write f(n) ≪ g(n); if the implied constant depends on a certain parameter c, then we write f(n) ≪c g(n). A, B,... denote subsets of N0 and A + B denotes the set of the nonnegative integers n that can be represented in the form n = a + b with a ∈ A, b ∈ B. ω(n) denotes the number of distinct prime factors of n and Ω(n) denotes the number of prime factors of n counted with multiplicity. λ(n) is the Liouville function: λ(n) = (−1) Ω(n). The divisor function is denoted by τ(n).
Multivariate Diophantine equations with many solutions
, 2001
"... Among other things we show that for each ntuple of positive rational numbers (a 1 ; : : : ; a n ) there are sets of primes S of arbitrarily large cardinality s such that the solutions of the equation a 1 x 1 + +a n x n = 1 with x 1 ; : : : ; x n Sunits are not contained in fewer than exp((4 + ..."
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Cited by 3 (1 self)
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Among other things we show that for each ntuple of positive rational numbers (a 1 ; : : : ; a n ) there are sets of primes S of arbitrarily large cardinality s such that the solutions of the equation a 1 x 1 + +a n x n = 1 with x 1 ; : : : ; x n Sunits are not contained in fewer than exp((4 + o(1))s 1=2 (log s) 1=2 ) proper linear subspaces of C n . This generalizes a result of Erdős, Stewart and Tijdeman [7] for Sunit equations in two variables. Further, we prove that for any algebraic number field K of degree n, any integer m with 1 m < n, and any sufficiently large s there are integers 0 ; : : : ; m in K which are linearly independent over Q , and prime numbers p 1 ; : : : ; p s , such that the norm polynomial equation jN K=Q ( 0 + 1 x 1 + + mxm )j = p z1 1 p zs s has at least expf(1+o(1)) n m s m=n (log s) 1+m=n g solutions in x 1 ; : : : ; xm ; z 1 ; : : : ; z s 2 Z. This generalizes a result of Moree and Stewart [19] for m = 1. Our main tool, also established in this paper, is an effective lower bound for the number K;T (X; Y ) of ideals in a number field K of norm X composed of prime ideals which lie outside a given finite set of prime ideals T and which have norm Y . This generalizes results of Caneld, Erdős and Pomerance [6] and of Moree and Stewart [19].
Two Sunit equations with many solutions
 J. Number Theory
, 2007
"... In this note we consider two Sunit equations for which we will exhibit many solutions. Our first problem concerns solutions to the equation a + b = c where a, b, and c are coprime integers such that all prime factors of abc lie in a given set S of s primes. In [8] J.H. Evertse showed that this Su ..."
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Cited by 2 (0 self)
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In this note we consider two Sunit equations for which we will exhibit many solutions. Our first problem concerns solutions to the equation a + b = c where a, b, and c are coprime integers such that all prime factors of abc lie in a given set S of s primes. In [8] J.H. Evertse showed that this Sunit equation has at most exp(4s + 6) solutions. On the other
The Number Of Solutions Of Diophantine Equations
"... Introduction. In two recent papers [4], [30], Erdos, Stewart and the author showed that certain diophantine equations have many solutions. In this way they indicated how far certain results are capable for improvements at most. First we mention some relevant results from the literature on upper boun ..."
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Introduction. In two recent papers [4], [30], Erdos, Stewart and the author showed that certain diophantine equations have many solutions. In this way they indicated how far certain results are capable for improvements at most. First we mention some relevant results from the literature on upper bounds for the numbers of solutions of diophantine equations and then we sketch how our method leads to opposite results. 1. Thue and ThueMahler equations. Let f(x; y) = a 0 x n + a 1 x n\Gamma1 y + ::: + an y n 2 Z [x ; y ] be a binary form (i.e. homogeneous polynomial) of degree n 3: Put A := max j=0;:::;n ja j j. Let m<F39.