Dirichlet's Approximation Theorem Research Articles

Diophantine approximation with prime denominator in real quadratic function fields

In the thirties of the last century, I.M. Vinogradov proved that the inequality ||pα||≤p−1/5+ε has infinitely prime solutions p, where ||.|| denotes the distance to a nearest integer. This result has subsequently been improved by many authors. In particular, Vaughan (1978) replaced the exponent 1/5 by 1/4 using his celebrated identity for the von Mangoldt function and a refinement of Fourier analytic arguments. The current record is due to Matomäki (2009) who showed the infinitude of prime solutions of the inequality ||pα||≤p−1/3+ε. This exponent 1/3 is considered the limit of the current technology. Recently, in [3], the authors established an analogue of Matomäki's result on imaginary quadratic extensions of the function field k=Fq(T). In this paper, we consider the case of real quadratic extensions of k of class number 1, for which we prove a function field analogue of Vaughan's above-mentioned result (exponent θ=1/4). Our method uses versions of Vaughan's identity and the Dirichlet approximation theorem for function fields. The latter was established by Arijit Ganguly in the appendix to our previous paper [3] on the imaginary quadratic case. We also simplify arguments in the paper [1] on the same problem for real quadratic number fields by D. Mazumder and the first-named author.

Limit distributions of expanding translates of shrinking submanifolds and non-improvability of Dirichlet's approximation theorem

Limit distributions of expanding translates of shrinking submanifolds and non-improvability of Dirichlet's approximation theorem

Diophantine approximation with prime restriction in function fields

In the thirties of the last century, I. M. Vinogradov established uniform distribution modulo 1 of the sequence pα when α is a fixed irrational real number and p runs over the primes. In particular, he showed that the inequality ||pα||≤p−1/5+ε has infinitely prime solutions p, where ||.|| denotes the distance to a nearest integer. This result has subsequently been improved by many authors. The current record is due to Matomäki (2009) who showed the infinitude of prime solutions of the inequality ||pα||≤p−1/3+ε. This exponent 1/3 is considered the limit of the current technology. We prove function field analogues of this result for the fields k=Fq(T) and imaginary quadratic extensions K of k. Essential in our method is the Dirichlet approximation theorem for function fields which is established in general form in the appendix authored by Arijit Ganguly.

The Minkowski chain and Diophantine approximation

The Hurwitz chain gives a sequence of pairs of Farey approximations to an irrational real number. Minkowski gave a criterion for a number to be algebraic by using a certain generalization of the Hurwitz chain. We apply Minkowski's generalization (the Minkowski chain) to give criteria for a real linear form to be either badly approximable or singular. We also give a variant of Dirichlet's approximation theorem for a real linear form that produces a whole basis of approximating integral vectors rather than a single one. This result holds if and only if the form is badly approximable. The proofs rely on properties of successive minima and reduced bases of lattices.

Equidistribution of curves in homogeneous spaces and Dirichlet's approximation theorem for matrices

In this paper, we study an analytic curve $ \varphi: I = [a, b]\rightarrow \mathrm{M}(m\times n, \mathbb{R}) $ in the space of $ m $ by $ n $ real matrices, and show that if $ \varphi $ satisfies certain geometric condition, then for almost every point on the curve, the Diophantine approximation given by Dirichlet's Theorem can not be improved. To do this, we embed the curve into a homogeneous space $ G/\Gamma $, and prove that under the action of some expanding diagonal subgroup $ A = \{a(t): t \in \mathbb{R}\} $, the translates of the curve tend to be equidistributed in $ G/\Gamma $, as $ t \rightarrow +\infty $. The proof relies on the linearization technique and representation theory.

The q-unit circle: The unit circle in prime characteristics and its properties

We define the unit circle for global function fields. We demonstrate that this unit circle, endearingly termed the q-unit circle (pronounced “cue-nit”), after the finite field Fq of q elements, enjoys all of the properties akin to the classical unit circle: center, curvature, roots of unity in completions, integrality conditions, embedding into a finite-dimensional vector space over the real line, a partition of the ambient space into concentric circles, Möbius transformations, a Dirichlet approximation theorem, a reciprocity law, and much more. In addition, we extend the polynomial exponential action of Carlitz to an action by all points on the real line; we show that mutually tangent horoballs solve a Descartes-type relation arising from reciprocity; we define the hyperbolic plane, which we prove is uniquely determined by the q-unit circle; and we give the associated modular forms and Eisenstein series.

Root geometry of polynomial sequences II: Type (1,0)

We consider the sequence of polynomials Wn(x) defined by the recursion Wn(x)=(ax+b)Wn−1(x)+dWn−2(x), with initial values W0(x)=1 and W1(x)=t(x−r), where a, b, d, t, r are real numbers, with a,t>0 and d<0. It is known that every polynomial Wn(x) is distinct-real-rooted. We find that, as n→∞, the smallest root of the polynomial Wn(x) converges decreasingly to a real number, and that the largest root converges increasingly to a real number. Moreover, by using the Dirichlet approximation theorem, we prove that for every integer j≥2, the jth smallest root of the polynomial Wn(x) converges as n→∞, and so does the jth largest root. It turns out that these two convergence points are independent of the numbers t, r, and the integer j. We obtain explicit expressions for the above four limit points.

A generalization of Dirichlet approximation theorem for the affine actions on real line

In this paper, we obtain strong density results for the orbits of real numbers under the action of the semigroup generated by the affine transformations T 0 ( x ) = x / a and T 1 ( x ) = b x + 1 , where a , b > 1 . These density results are formulated as generalizations of the Dirichlet approximation theorem and improve the results of Bergelson, Misiurewicz, and Senti. We show that for any x , u > 0 there are infinitely many elements γ in the semigroup generated by T 0 and T 1 such that | γ ( x ) − u | < C ( t 1 / | γ | − 1 ) , where C and t are constants independent of γ, and | γ | is the length of γ as a word in the semigroup. Finally, we discuss the problem of approximating an arbitrary real number by the ratios of prime numbers and the ratios of logarithms of prime numbers.

A note on Hornich's problem

This note, by applying Dirichlet's approximation theorem, achieves an upper bound for Hornich's problem, the problem to determine the smallest number of nonoverlapping unit spheres which can block all the light rays starting from the center of a fixed one.

On Dirichlet's approximation theorem

Though we cannot improve on the upper bound in Dirichlet's approximation theorem,Kaindl has shown that the upper bound can be lowered fromtn totn−tn−1−tn−2−...−t−1, if we admit equality. We show thatKaindl's upper bound is lowest possible in this case. The result is then generalized to linear forms.

Zum Approximationssatz von Dirichlet

In this paper we improve the estimation in Dirichlet's Approximation Theorem for systems of linear forms using a method ofG. Raisbeck andG. Kaindl.

Two questions in diophantine approximation

The first question is about a possible variation on Dirichlet's approximation theorem for linear forms αx+βy+z, wherex,y are restricted topositive integers. The second question, which turns out to be related to the first, is about approximation to elements in a power series fieldk((t −1)) by solutions β of first order linear differential equationsxβ′+yβ+z=0, wherex,y,z are polynomials int.

A feature of Dirichlet's approximation theorem

A feature of Dirichlet's approximation theorem

A Note on the Distribution of the Zeros of the Zeta-Function

on the other hand, where pi = pi + iy1, P2 = /2 + iy2, . . . denote. the zeros of g(s) in the upper half-plane. While for the purposes of Littlewood only a consequence of the Dirichlet approximation theorem is relevant, it seems to be worth while to determine the asymptotic cyclical distribution of (1). The object of the present note is to point out the fact that the sequence (1) is asymptotically equidistributed t to modulus c, where c is any non-vanishing real number. In order to simplify the formulae we shall choose c = 1; it will be clear that the proof holds for any c. We have to prove that