A Generalization of Furstenberg’s Diophantine Theorem
نویسندگان
چکیده
We obtain a generalization of Furstenberg’s Diophantine Theorem on non-lacunary multiplicative semigroups. For example we show that the sets of sums {(p1 qm 1 + p2 qm 2 )α : n, m ∈ N} and {(p1 qm 1 + 2n)α : n, m ∈ N} are dense in the circle T = R/Z for all irrational α, where (pi, qi) are distinct pairs of multiplicatively independent integers for i = 1, 2.
منابع مشابه
VIP systems in partial semigroups
A VIP system is a polynomial type generalization of the notion of an IP system, i.e., a set of finite sums. We extend the notion of VIP system to commutative partial semigroups and obtain an analogue of Furstenberg’s central sets theorem for these systems which extends the polynomial Hales-Jewett Theorem of Bergelson and Leibman. Several Ramsey Theoretic consequences, including the central sets...
متن کاملSelf-similar fractals and arithmetic dynamics
The concept of self-similarity on subsets of algebraic varieties is defined by considering algebraic endomorphisms of the variety as `similarity' maps. Self-similar fractals are subsets of algebraic varieties which can be written as a finite and disjoint union of `similar' copies. Fractals provide a framework in which, one can unite some results and conjectures in Diophantine g...
متن کاملA Generalized Fibonacci Sequence and the Diophantine Equations $x^2pm kxy-y^2pm x=0$
In this paper some properties of a generalization of Fibonacci sequence are investigated. Then we solve the Diophantine equations $x^2pmkxy-y^2pm x=0$, where $k$ is positive integer, and describe the structure of solutions.
متن کاملA Generalization of Stanley’s Monster Reciprocity Theorem
By studying the reciprocity property of linear Diophantine systems in light of Malcev-Neumann series, we present in this paper a new approach to and a generalization of Stanley’s monster reciprocity theorem. A formula for the “error term” is given in the case when the system does not have the reciprocity property. We also give a short proof of Stanley’s reciprocity theorem for linear homogeneou...
متن کاملSome remarks on diophantine equations and diophantine approximation
We give many equivalent statements of Mahler’s generalization of the fundamental theorem of Thue. In particular, we show that the theorem of Thue–Mahler for degree 3 implies the theorem of Thue for arbitrary degree ≥ 3, and we relate it with a theorem of Siegel on the rational integral points on the projective line P(K) minus 3 points. Classification MSC 2010: 11D59; 11J87; 11D25
متن کامل