A note on configurations in sets of positive density which occur at all large scales
Morris, ID (2015) A note on configurations in sets of positive density which occur at all large scales Israel Journal of Mathematics, 207 (2). pp. 719738.
Full text not available from this repository.Abstract
© 2015, Hebrew University of Jerusalem.Furstenberg, Katznelson and Weiss proved in the early 1980s that every measurable subset of the plane with positive density at infinity has the property that all sufficiently large real numbers are realised as the Euclidean distance between points in that set. Their proof used ergodic theory to study translations on a space of Lipschitz functions corresponding to closed subsets of the plane, combined with a measuretheoretical argument. We consider an alternative dynamical approach in which the phase space is given by the set of measurable functions from ℝ^{d} to [0, 1], which we view as a compact subspace of L^{∞}(ℝ^{d}) in the weak* topology. The pointwise ergodic theorem for ℝ^{d}actions implies that with respect to any translationinvariant measure on this space, almost every function is asymptotically close to a constant function at large scales. This observation leads to a general sufficient condition for a configuration to occur in every set of positive upper Banach density at all sufficiently large scales. To illustrate the use of this criterion we apply it to prove a new result concerning threepoint configurations in measurable subsets of the plane which form the vertices of a triangle with specified area and side length, yielding a new proof of a result related to work of R. Graham.
Item Type:  Article  

Divisions :  Faculty of Engineering and Physical Sciences > Mathematics  
Authors : 


Date :  28 March 2015  
DOI :  10.1007/s1185601511873  
Depositing User :  Symplectic Elements  
Date Deposited :  17 May 2017 13:39  
Last Modified :  10 Jun 2019 13:12  
URI:  http://epubs.surrey.ac.uk/id/eprint/839994 
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