There are two main interrelated goals of this paper. Firstly we investigate the sums

Abstract. In [7] the authors set out a programme to prove the Duffin-Schaeffer Conjecture for measures arbitrarily close to Lebesgue measure. In this paper we take a new step in this direction. Given a nonnegative function ψ : N → R, let W (ψ) denote the set of real numbers x such that |nx − a| < ψ(n) for infinitely many reduced rationals a/n (n > 0). Our main result is that W (ψ) is of full Lebesgue measure if there exists a c > 0 such that ∑ n≥16 ϕ(n)ψ(n) n exp(c(log log n)(log log log n)) = ∞ .

Abstract. Given a nonnegative function ψ : N → R, let W (ψ) denote the set of real numbers x such that |nx − a| < ψ(n) for infinitely many reduced rationals a/n (n > 0). A consequence of our main result is that W (ψ) is of full Lebesgue measure if there exists an ǫ > 0 such thatThe Duffin-Schaeffer Conjecture is the corresponding statement with ǫ = 0 and represents a fundamental unsolved problem in metric number theory. Another consequence is that W (ψ) is of full Hausdorff dimension if the above sum with ǫ = 0 diverges; i.e. the dimension analogue of the Duffin-Schaeffer Conjecture is true.

In 1998, Burago–Kleiner and McMullen independently proved the existence of separated nets in double-struckRd which are not bi‐Lipschitz equivalent (BL) to a lattice. A finer equivalence relation than BL is bounded displacement (BD). Separated nets arise naturally as return times to a section for minimal double-struckRd‐actions. We analyze the separated nets which arise via these constructions, focusing particularly on nets arising from linear double-struckRd‐actions on tori. We show that generically these nets are BL to a lattice, and for some choices of dimensions and sections, they are generically BD to a lattice. We also show the existence of such nets which are not BD to a lattice.

For the development of a mathematical theory which can be used to rigorously investigate physical properties of quasicrystals, it is necessary to understand regularity of patterns in special classes of aperiodic point sets in Euclidean space. In one dimension, prototypical mathematical models for quasicrystals are provided by Sturmian sequences and by point sets generated by substitution rules. Regularity properties of such sets are well understood, thanks mostly to well known results by Morse and Hedlund, and physicists have used this understanding to study one dimensional random Schrödinger operators and lattice gas models. A key fact which plays an important role in these problems is the existence of a subadditive ergodic theorem, which is guaranteed when the corresponding point set is linearly repetitive.In this paper we extend the one-dimensional model to cut and project sets, which generalize Sturmian sequences in higher dimensions, and which are frequently used in mathematical and physical literature as models for higher dimensional quasicrystals. By using a combination of algebraic, geometric, and dynamical techniques, together with input from higher dimensional Diophantine approximation, we give a complete characterization of all linearly repetitive cut and project sets with cubical windows. We also prove London Mathematical Society* Research supported by EPSRC grants EP/L001462, EP/J00149X, EP/M023540. HK also gratefully acknowledges the support of the Osk. Huttunen foundation.