Fourier transforms of Gibbs measures for the Gauss map

We investigate under which conditions a given invariant measure $\mu$ for the dynamical system defined by the Gauss map $x \mapsto 1/x \mod 1$ is a Rajchman measure with polynomially decaying Fourier transform $$|\widehat{\mu}(\xi)| = O(|\xi|^{-\eta}), \quad \text{as } |\xi| \to \infty.$$ We show that this property holds for any Gibbs measure $\mu$ of Hausdorff dimension greater than $1/2$ with a natural large deviation assumption on the Gibbs potential. In particular, we obtain the result for the Hausdorff measure and all Gibbs measures of dimension greater than $1/2$ on badly approximable numbers, which extends the constructions of Kaufman and Queff\'elec-Ramar\'e. Our main result implies that the Fourier-Stieltjes coefficients of the Minkowski's question mark function decay to $0$ polynomially answering a question of Salem from 1943. As an application of the Davenport-Erd\H{o}s-LeVeque criterion we obtain an equidistribution theorem for Gibbs measures, which extends in part a recent result by Hochman-Shmerkin. Our proofs are based on exploiting the nonlinear and number theoretic nature of the Gauss map and large deviation theory for Hausdorff dimension and Lyapunov exponents.Comment: v3: 29 pages; peer-reviewed version, fixes typos and added more elaborations, and included comments on Salem's problem. To appear in Math. An

An analysis of regional commuting flows in the European Union

Regional labour mobility is of increasing concern in the context of the Single European Monetary Policy, as EMU implies a reduction of national policy options. Thus, it is important that the remaining adjustment mechanisms function effectively. While most of the empirical literature focuses on labour mobility in terms of migration, this paper provides an empirical assessment of the determinants of aggregate regional commuting flows in the EU, an issue often examined in a local or national context but still un(der)explored on EU level. Using an extended gravity framework, commuting is found to respond to differences in regional wages and unemployment, and to provide an equilibrating mechanism to labour market disequilibria. Higher levels of education and labour force participation of women, as well as a larger services sector are associated with a higher percentage of commuting. Finally, the results reveal interesting geographical differences between internal, border and coastal regions.Labour mobility, regional commuting, EMU, gravity model