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On the Multiplicative Group Generated by Two Primes in Z∕QZ
Abstract
We study the action of the multiplicative group generated by two prime numbers in Z∕Q
Z. More specifically, we study returns to the set ([−Qε, Qε] ∩Z)∕Q
Z. This is intimately related to the problem of bounding the greatest common divisor of S-unit differences, which we revisit. Our main tool is the S-adic subspace theorem.
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We give results and inequalities bounding the greatest common divisor of multivariable polynomials evaluated at S-unit arguments, generalizing to an arbitrary number of variables results of Bugeaud–Corvaja–Zannier, Hernández–Luca, and Corvaja–Zannier. In closely related results, and in line with observations of Silverman, we prove special cases of Vojta’s conjecture for blowups of toric varieties. As an application, we classify when terms from simple linear recurrence sequences can have a large greatest common divisor (in an appropriate sense). The primary tool used in the proofs is Schmidt’s Subspace Theorem from Diophantine approximation.
For a nonsingular integer matrix A, we study the growth of the order of A modulo N. We say that a matrix is exceptional if it is diagonalizable, and a power of the matrix has all eigenvalues equal to powers of a single rational integer, or all eigenvalues are powers of a single unit in a real quadratic field. For exceptional matrices, it is easily seen that there are arbitrarily large values of N for which the order of A modulo N is logarithmically small. In contrast, we show that if the matrix is not exceptional, then the order of A modulo N goes to infinity faster than any constant multiple of log N. Comment: Added references and corrected a few misprints. Added condition that A be ergodic for a remark in the introduction
This book considers the so-called unlikely intersections, a topic that embraces well-known issues, such as Lang's and Manin–Mumford's, concerning torsion points in subvarieties of tori or abelian varieties. More generally, the book considers algebraic subgroups that meet a given subvariety in a set of unlikely dimension. The book is an expansion of the Hermann Weyl Lectures delivered by the author at the Institute for Advanced Study in Princeton in May 2010.The book consists of four chapters and seven brief appendixes. The first chapter considers multiplicative algebraic groups, presenting proofs of several developments, ranging from the origins to recent results, and discussing many applications and relations with other contexts. The second chapter considers an analogue in arithmetic and several applications of this. The third chapter introduces a new method for approaching some of these questions, and presents a detailed application of this to a relative case of the Manin–Mumford issue. The fourth chapter focuses on the André–Oort conjecture (outlining work by Pila).
We prove that for integers a > b > c > 0 a>b>c>0 , the greatest prime factor of ( a b + 1 ) ( a c + 1 ) (ab+1)(ac+1) tends to infinity with a a . In particular, this settles a conjecture raised by Györy, Sarkozy and Stewart, predicting the same conclusion for the product ( a b + 1 ) ( a c + 1 ) ( b c + 1 ) (ab+1)(ac+1)(bc+1) .
Applications of Diophantine Approximation to Integral Points and Transcendence - by Pietro Corvaja May 2018
Cambridge Core - Geometry and Topology - Applications of Diophantine Approximation to Integral Points and Transcendence - by Pietro Corvaja
This book considers the so-called Unlikely Intersections, a topic that embraces well-known issues, such as Lang's and Manin-Mumford's, concerning torsion points in subvarieties of tori or abelian varieties. More generally, the book considers algebraic subgroups that meet a given subvariety in a set of unlikely dimension. The book is an expansion of the Hermann Weyl Lectures delivered by Umberto Zannier at the Institute for Advanced Study in Princeton in May 2010. The book consists of four chapters and seven brief appendixes, the last six by David Masser. The first chapter considers multiplicative algebraic groups, presenting proofs of several developments, ranging from the origins to recent results, and discussing many applications and relations with other contexts. The second chapter considers an analogue in arithmetic and several applications of this. The third chapter introduces a new method for approaching some of these questions, and presents a detailed application of this (by Masser and the author) to a relative case of the Manin-Mumford issue. The fourth chapter focuses on the André-Oort conjecture (outlining work by Pila).
Diophantine geometry has been studied by number theorists for thousands of years, since the time of Pythagoras, and has continued to be a rich area of ideas such as Fermat's Last Theorem, and most recently the ABC conjecture. This monograph is a bridge between the classical theory and modern approach via arithmetic geometry. The authors provide a clear path through the subject for graduate students and researchers. They have re-examined many results and much of the literature, and provide a thorough account of several topics at a level not seen before in book form. The treatment is largely self-contained, with proofs given in full detail. © Cambridge University Press 2006 and Cambridge University Press, 2009.
We prove a more general form of a conjecture of Gyory, Sarkozy and Stewart concerning the largest prime factor of expressions of the form (ab + 1)(ac + 1)(bc + 1) with distinct positive integers a, b, c.
In [2], it was shown that if a and b are multiplicatively independent integers and ɛ > 0, then the inequality gcd (a
n
− 1,b
n
− 1) < exp(ɛn) holds for all but finitely many positive integers n. Here, we generalize the above result. In particular, we show that if f(x),f
1(x),g(x),g
1(x) are non-zero polynomials with integer coefficients, then for every ɛ > 0, the inequality
holds for all but finitely many positive integers n.
Let a,b be given, multiplicatively independent positive integers and let ε>0. In a recent paper jointly with Y. Bugeaud we proved the upper bound exp(εn) for g.c.d.(a
n
−1, b
n
−1); shortly afterwards we generalized this to the estimate g.c.d.(u−1,v−1)<max(∣u∣,∣v∣)ε for multiplicatively independent S-units u,v∈Z. In a subsequent analysis of those results it turned out that a perhaps better formulation of them may be obtained in terms of the language of heights of algebraic numbers. In fact, the purposes of the present paper are: to generalize the upper bound for the g.c.d. to pairs of rational functions other than {u−1,v−1} and to extend the results to the realm of algebraic numbers, giving at the same time a new formulation of the bounds in terms of height functions and algebraic subgroups of G
m2.