After defining a notion of epsilon-density, we provide for any integer m>1 and real algebraic number alpha an estimate of the smallest epsilon such that the set of vectors of the form (t,t^alpha,...,t alpha^{m-1}) for tR is epsilon-dense modulo 1 in terms of the multiplicative Mahler measure M(A(x)) of the minimal integral polynomial A(x) of alpha, which is independent of m. In particular, we show that if alpha has degree d it is possible to take epsilon = 2^{[d/2]}/M(A(x)). On the other side we show using asymptotic estimates for Toeplitz determinants that we cannot have epsilon$-density for sufficiently large m whenever epsilon is strictly smaller than 1/M(A(x)). In the process of proving this we obtain a result of independent interest about the structure of the Z-module of integral linear recurrences of fixed length determined by a non-monic polynomial.

%B Acta Arithmetica %V 153 %P 15-33 %G eng %U http://journals.impan.pl/cgi-bin/doi?aa153-1-2 %N 1 %R 10.4064/aa153-1-2 %0 Conference Paper %B 2011 Congress of the International Federation of Automatic Control - IFAC2011 %D 2011 %T Left invertibility of output-quantized MISO linear systems %A N. Dubbini %A M. Monge %A A. Bicchi %K Embedded Control %K Robotics %XThis paper studies left invertibility of single-output discrete-time quantized linear systems. Quantized outputs are generated according to a given partition of the state-space, while inputs are sequences on a finite alphabet. Left invertibility deals with the possibility of recovering unknown inputs from the only knowledge of the outputs. It is reduced, under suitable conditions, to left D-invertibility: while left invertibility takes into account membership to sets of a given partition, left D-invertibility considers only membership to a single set, and is easily (and algorithmically) detectable. Our main result is a sufficient condition for the equivalence between left invertibility and left D-invertibility in MISO system. In unidimensional systems the equivalence is valid except at most a finite (and computable) number of cases. These results allows the effective detection of left invertibility by means of left Dinvertibility, which is algorithmically detectable. An example with effective computations is presented to show the application of the proposed method.

%B 2011 Congress of the International Federation of Automatic Control - IFAC2011 %C Milano, italy %P 11278 - 11283 %8 August 28 - Sept %G eng