A Hausdorff measure version of the Duffin-Schaeffer conjecture in metric number theory is introduced and discussed. The general conjecture is established modulo the original conjecture. The key result is a Mass Transference Principle which allows us to transfer Lebesgue measure theoretic statements for lim sup subsets of R k to Hausdorff measure theoretic statements. In view of this, the Lebesgue theory of lim sup sets is shown to underpin the general Hausdorff theory. This is rather surprising since the latter theory is viewed to be a subtle refinement of the former.

A mass transference principle and the Duffin-Schaeffer conjecture for Hausdorff measures

A Hausdorff measure version of the Duffin-Schaeffer conjecture in metric number theory is introduced and discussed. The general conjecture is established modulo the original conjecture. The key result is a Mass Transference Principle which allows us to transfer Lebesgue measure theoretic statements for lim sup subsets of R^k to Hausdorff measure theoretic statements. In view of this, the Lebesgue theory of lim sup sets is shown to underpin the general Hausdorff theory. This is rather surprising since the latter theory is viewed to be a subtle refinement of the former.

A Mass Transference Principle and the Duffin-Schaeffer conjecture for Hausdorff measures

Let ([0, 1), T) be the dynamical system of continued fractions. Let {zn}n⩾1 be a sequence of real numbers in [0, 1] and ψ: ℕ × [0, 1) → ℝ+ be a positive function. A point x∈[0, 1) is said to be ψ‐approximable by {zn}n ⩾ 1 if |Tnx − zn| < ψ(n, x) holds for infinitely many n ∈ ℕ. In this paper, the Hausdorff dimension of the set of ψ‐approximable points is studied. The dimensions are completely determined when ψ(n, x) = ψ(n) independent on x and when ψ(n,x)=e−(f(x)+⋯+f(Tn−1x)) with f a positive continuous function. For the proof of these results, a relationship between a ball in [0, 1) and the cylinders defined by the partial quotients in continued fractions is investigated. It is shown that a ball can be sufficiently packed by cylinders of the same order and of comparable length, which gives us explicit continued fraction representations in locating the points in a ball in [0, 1).

The shrinking target problem in the dynamical system of continued fractions

The idea of using measure theoretic concepts to investigate the size of number theoretic sets, originating with E. Borel, has been used for nearly a century. It has led to the development of the theory of metrical Diophantine approximation, a branch of Number Theory which draws on a rich and broad variety of mathematics. We discuss some recent progress and open problems concerning this classical theory. In particular, generalisations of the Duffin-Schaeffer and Catlin conjectures are formulated and explored.

Classical metric Diophantine approximation revisited

Let denote the set of ψ-approximable points in . Under the assumption that the approximating function ψ is monotonic, the classical Khintchine-Groshev theorem provides an elegant probabilistic criterion for the Lebesgue measure of . The famous Duffin-Schaeffer counterexample shows that the monotonicity assumption on ψ is absolutely necessary when m = n = 1. On the other hand, it is known that monotonicity is not necessary when n ≥ 3 (Schmidt) or when n = 1 and m ≥ 2 (Gallagher). Surprisingly, when n = 2, the situation is unresolved. We deal with this remaining case and thereby remove all unnecessary conditions from the classical Khintchine-Groshev theorem. This settles a multidimensional analog of Catlin's conjecture.

Classical Metric Diophantine Approximation Revisited: The Khintchine–Groshev Theorem

Given a compact metric space (X,d) equipped with a non-atomic, probability measure m and a real, positive decreasing function p we consider a `natural' class of limsup subsets La(p) of X. The classical limsup sets of `well approximable' numbers in the theory of metric Diophantine approximation fall within this class. We show that m(La(p))>0 under a `global ubiquity' hypothesis and the divergence of a certain m--volume sum. In fact, under a `local ubiquity' hypothesis we show that La(p) has full measure; i.e. m(La(p)) =1 . This is the analogue of the divergent part of the classical Khintchine-Groshev theorem in number theory. Moreover, if the 'local ubiquity' hypothesis is satisfied and a certain f-volume sum diverges then we are able to show that the Hausdorff f--measure of La(p) is infinite. A simple consequence of this is a lower bound for the Hausdorff dimension of La(p) and various results concerning the dimension and measure of related `exact order' sets. 
Essentially, the notion of `local ubiquity' unexpectedly unifies `divergent' type results for La(p) with respect to the natural measure m and general Hausdorff measures. Applications of the general framework include those from number theory, Kleinian groups and rational maps. Even for the classical limsup sets of `well approximable' numbers, the framework strengthens the classical Hausdorff measure result of Jarnik and opens up the Duffin-Schaeffer conjecture for Hausdorff measures.

Measure theoretic laws for lim sup sets

Let C be a nondegenerate planar curve and for a real, positive decreasing function ψ let C(ψ) denote the set of simultaneously ψ-approximable points lying on C. We show that C is of Khintchine type for divergence; i.e. if a certain sum diverges then the one-dimensional Lebesgue measure on C of C(ψ) is full. We also obtain the Hausdorff measure analogue of the divergent Khintchine type result. In the case that C is a rational quadric the convergence counterparts of the divergent results are also obtained. Furthermore, for functions ψ with lower order in a critical range we determine a general, exact formula for the Hausdorff dimension of C(ψ). These results constitute the first precise and general results in the theory of simultaneous Diophantine approximation on manifolds.

Diophantine approximation onplanar curves andthe distribution of rational points

Hausdorff measure and linear forms.

Let
p
be a prime number. For a positive integer
n
and a
p
-adic number
ξ
, let
λ
n
(
ξ
) denote
the supremum of the real numbers
λ
such that there are arbitrarily large positive integers
q
such that
qξ
p
,
qξ
2
p
,...,
qξ
n
p
are all less than
q
−
λ
−
1
. Here,
x
p
denotes the infimum of
|
x
−
n
|
p
as
n
runs
through the integers. We study the set of values taken by the function
λ
n

/pdf/on-simultaneous-rational-approximation-to-a-p-adic-number-c03kmhgsli.pdf

On simultaneous rational approximation to a p-adic number and its integral powers

APPROXIMATION BY ALGEBRAIC NUMBERS (Cambridge Tracts in Mathematics 160) By YANN BUGEAUD: 274 pp., £40.00 (US$70.00), ISBN 0-521-82329-3 (Cambridge University Press, 2004)

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