Diophantine Analysis and Additive Theory
نویسنده
چکیده
A summary of research papers presented for the Habilitationsgesuch Preface In total, 11 papers are presented for the Habilitationsgesuch. They can be divided into three big topics. The results of these papers are shortly described below. In this text, I sometimes allow myself to be slightly sloppy in definitions and formulations, to avoid unnecessary technicalities. The papers listed above are cited as [A1], [A2], [B1], etc. The other papers (listed in the bibliography below) are cited as [1], [2], etc.
منابع مشابه
Diophantine Problems in Many Variables: the Role of Additive Number Theory
We provide an account of the current state of knowledge concerning diophantine problems in many variables, paying attention in particular to the fundamental role played by additive number theory in establishing a large part of this body of knowledge. We describe recent explicit versions of the theorems of Brauer and Birch concerning the solubility of systems of forms in many variables, and esta...
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We consider Euclid’s gcd algorithm for two integers (p, q) with 1 ≤ p ≤ q ≤ N , with the uniform distribution on input pairs. We study the distribution of the total cost of execution of the algorithm for an additive cost function d on the set of possible digits, asymptotically for N → ∞. For any additive cost of moderate growth d, Baladi and Vallée obtained a central limit theorem, and –in the ...
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We prove a strong simultaneous Diophantine approximation theorem for values of additive and multiplicative functions provided that the functions have certain regularity on the primes.
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Diophantine Analysis is a very active domain of mathematical research where one finds more conjectures than results. We collect here a number of open questions concerning Diophantine equations (including Pillai’s Conjectures), Diophantine approximation (featuring the abc Conjecture) and transcendental number theory (with, for instance, Schanuel’s Conjecture). Some questions related to Mahler’s ...
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For large N , we consider the ordinary continued fraction of x = p/q with 1 ≤ p ≤ q ≤ N , or, equivalently, Euclid’s gcd algorithm for two integers 1 ≤ p ≤ q ≤ N , putting the uniform distribution on the set of p and qs. We study the distribution of the total cost of execution of the algorithm for an additive cost function c on the set Z∗+ of possible digits, asymptotically for N → ∞. If c is n...
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