On the dimension of iterated sumsets

Let A be a subset of the real line. We study the fractal dimensions of the k-fold iterated sumsets kA, defined as kA = A+...+A (k times). We show that for any non-decreasing sequence {a_k} taking values in [0,1], there exists a compact set A such that kA has Hausdorff dimension a_k for all k. We also show how to control various kinds of dimension simultaneously for families of iterated sumsets. These results are in stark contrast to the Plunnecke-Rusza inequalities in additive combinatorics. However, for lower box-counting dimension, the analogue of the Plunnecke-Rusza inequalities does hold.Comment: To appear in the Proceedings of the Conference on Fractals and Related Fields, Monastir, 200

Expected Inflation, Expected Stock Returns, and Money Illusion: What can we learn from Survey Expectations?

We show empirically that survey-based measures of expected inflation are significant and strong predictors of future aggregate stock returns in several industrialized countries both in-sample and out-of-sample. By empirically discriminating between competing sources of this return predictability by virtue of a comprehensive set of expectations data, we find that money illusion seems to be the driving force behind our results. Another popular hypothesis - inflation as a proxy for aggregate risk aversion - is not supported by the data.Inflation expectations, Money Illusion, Proxy hypothesis, Stock returns

Further Baire results on the distribution of subsequences

This paper presents results about the distribution of subsequences which are typical in the sense of Baire. The first part is concerned with sequences of the type x_k = n_k*alpha, n_1 < n_2 < n_3 < ..., mod 1. Improving a result of Salat we show that, if the quotients q_k = n_{k+1}/n_k satisfy q_k > 1+ epsilon, then the set of alpha such that (x_k) is uniformly distributed is of first Baire category, i.e. for generic alpha we do not have uniform distribution. Under the stronger assumption lim q_k = infinity one even has maldistribution for generic alpha, the strongest possible contrast to uniform distribution. The second part reverses the point of view by considering appropriately defined Baire spaces S of subsequences. For a fixed well distributed sequence (x_n) we show that there is a set M of measures such that for generic (n_k) in S the set of limit measures of the subsequence (x_{n_k}) is exactly M.Comment: 21 pages, LaTeX2e. Final version. (Somewhat expanded proofs and clarifications, more examples

Dimension and product structure of hyperbolic measures

We prove that every hyperbolic measure invariant under a C^{1+\alpha} diffeomorphism of a smooth Riemannian manifold possesses asymptotically ``almost'' local product structure, i.e., its density can be approximated by the product of the densities on stable and unstable manifolds up to small exponentials. This has not been known even for measures supported on locally maximal hyperbolic sets. Using this property of hyperbolic measures we prove the long-standing Eckmann-Ruelle conjecture in dimension theory of smooth dynamical systems: the pointwise dimension of every hyperbolic measure invariant under a C^{1+\alpha} diffeomorphism exists almost everywhere. This implies the crucial fact that virtually all the characteristics of dimension type of the measure (including the Hausdorff dimension, box dimension, and information dimension) coincide. This provides the rigorous mathematical justification of the concept of fractal dimension for hyperbolic measures.Comment: 29 pages, published versio

On the Fourier dimension and a modification

We give a sufficient condition for the Fourier dimension of a countable union of sets to equal the supremum of the Fourier dimensions of the sets in the union, and show by example that the Fourier dimension is not countably stable in general. A natural approach to finite stability of the Fourier dimension for sets would be to try to prove that the Fourier dimension for measures is finitely stable, but we give an example showing that it is not in general. We also describe some situations where the Fourier dimension for measures is stable or is stable for all but one value of some parameter. Finally we propose a way of modifying the definition of the Fourier dimension so that it becomes countably stable, and show that a measure has modified Fourier dimension greater than or equal to $s$ if and only if it annihilates all sets with modified Fourier dimension less than $s$.Comment: v2: Added some remarks in the introduction and after Example 6. v3: Revised the introduction, strengthened Lemma 6, added Proposition 5 and Example 8. To appear in Journal of Fractal Geometr

Multifractal Analysis of Multiple Ergodic Averages

In this paper we present a complete solution to the problem of multifractal analysis of multiple ergodic averages in the case of symbolic dynamics for functions of two variables depending on the first coordinate.Comment: 5 pages, to appear in Comptes Rendus Mathematiqu

Automating Exchange Rate Target Zones: Intervention via an Electronic Limit Order Book

This paper describes and analyzes “automated intervention” of a target zone. Unusually detailed information about the order book allows studying intervention effects in a microstructure approach. We find in our sample that intervention increases exchange rate volatility (and spread) for the next minutes but that intervention days show a lower degree of volatility (and spread) than non-intervention days. We also show for intraday data that the price impact of interbank order flow is smaller on intervention days than on non-intervention days. Finally, we reveal that informed banks take different positions than uninformed banks as they tend to trade against the central bank – which reflects a rational stance. Despite this position taking, the targeted exchange rate range holds and volatility, spread and price impact go down. Overall, the credible expression of an intervention band seems to achieve the desired effects of a target zone.foreign exchange, microstructure, intervention, exchange rate