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116
A Construction of a SpaceTime Code Based on Number Theory
 IEEE Trans. Inform. Theory
, 2002
"... We construct a full data rate spacetime block code over M =2 transmit antennas and T =2 symbol periods, and we prove that it achieves a transmit diversity of 2 over all constellations carved from Z[i] . Further, we optimize the coding gain of the proposed code and then compare it to the Alamouti co ..."
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Cited by 62 (2 self)
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We construct a full data rate spacetime block code over M =2 transmit antennas and T =2 symbol periods, and we prove that it achieves a transmit diversity of 2 over all constellations carved from Z[i] . Further, we optimize the coding gain of the proposed code and then compare it to the Alamouti code. It is shown that the new code outperforms the Alamouti code at low and high SNR when the number of receive antennas N>1. The performance improvement is further enhanced when N or the size of the constellation increases. We relate the problem of spacetime diversity gain to algebraic number theory, and the coding gain optimization to the theory of simultaneous Diophantine approximation in the geometry of numbers. We find that the coding gain optimization is equivalent to find irrational numbers "the furthest" from any simultaneous rational approximations.
Uniform Spectral Properties Of OneDimensional Quasicrystals, IV. QuasiSturmian Potentials
 I. Absence of eigenvalues, Commun. Math. Phys
, 2000
"... We consider discrete onedimensional Schrodinger operators with quasiSturmian potentials. We present a new approach to the trace map dynamical system which is independent of the initial conditions and establish a characterization of the spectrum in terms of bounded trace map orbits. Using this, ..."
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Cited by 51 (32 self)
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We consider discrete onedimensional Schrodinger operators with quasiSturmian potentials. We present a new approach to the trace map dynamical system which is independent of the initial conditions and establish a characterization of the spectrum in terms of bounded trace map orbits. Using this, it is shown that the operators have purely singular continuous spectrum and their spectrum is a Cantor set of Lebesgue measure zero. We also exhibit a subclass having purely ffcontinuous spectrum. All these results hold uniformly on the hull generated by a given potential.
Random Generators and Normal Numbers
 EXPERIMENTAL MATHEMATICS
, 2000
"... Pursuant to the authors' previous chaoticdynamical model for random digits of fundamental constants [3], we investigate a complementary, statistical picture in which pseudorandom number generators (PRNGs) are central. Some rigorous results such as the following are achieved: Whereas the fundamental ..."
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Cited by 25 (11 self)
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Pursuant to the authors' previous chaoticdynamical model for random digits of fundamental constants [3], we investigate a complementary, statistical picture in which pseudorandom number generators (PRNGs) are central. Some rigorous results such as the following are achieved: Whereas the fundamental constant log 2 = P n2Z + 1=(n2 n ) is not yet known to be 2normal (i.e. normal to base 2), we are able to establish bnormality (and transcendency) for constants of the form P 1=(nb n ) but with the index n constrained to run over certain subsets of Z + . In this way we demonstrate, for example, that the constant 2;3 = P n=3;3 2 ;3 3 ;::: 1=(n2 n ) is 2normal. The constants share with ; log 2 and others the property that isolated digits can be directly calculated, but for the new class such computation is extraordinarily rapid. For example, we find that the googolth (i.e. 10 100  th) binary bit of 2;3 is 0. We also present a collection of other results  such as density results and irrationality proofs based on PRNG ideas  for various special numbers.
Euclidean algorithms are Gaussian
, 2003
"... Abstract. We prove a Central Limit Theorem for a general class of costparameters associated to the three standard Euclidean algorithms, with optimal speed of convergence, and error terms for the mean and variance. For the most basic parameter of the algorithms, the number of steps, we go further an ..."
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Cited by 22 (10 self)
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Abstract. We prove a Central Limit Theorem for a general class of costparameters associated to the three standard Euclidean algorithms, with optimal speed of convergence, and error terms for the mean and variance. For the most basic parameter of the algorithms, the number of steps, we go further and prove a Local Limit Theorem (LLT), with speed of convergence O((log N) −1/4+ǫ). This extends and improves the LLT obtained by Hensley [27] in the case of the standard Euclidean algorithm. We use a “dynamical analysis ” methodology, viewing an algorithm as a dynamical system (restricted to rational inputs), and combining tools imported from dynamics, such as the crucial transfer operators, with various other techniques: Dirichlet series, Perron’s formula, quasipowers theorems, the saddle point method. Dynamical analysis had previously been used to perform averagecase analysis of algorithms. For the present (dynamical) analysis in distribution, we require precise estimates on the transfer operators, when a parameter varies along vertical lines in the complex plane. Such estimates build on results obtained only recently by Dolgopyat in the context of continuoustime dynamics [20]. 1.
Controlling Strong Scarring for Quantized Ergodic Toral Automorphisms
, 2002
"... We show that in the semiclassical limit the eigenfunctions of quantized ergodic symplectic toral automorphisms can not concentrate in measure on a nite number of closed orbits of the dynamics. More generally, we show that, if the pure point component of the limit measure has support on a nite ..."
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Cited by 18 (0 self)
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We show that in the semiclassical limit the eigenfunctions of quantized ergodic symplectic toral automorphisms can not concentrate in measure on a nite number of closed orbits of the dynamics. More generally, we show that, if the pure point component of the limit measure has support on a nite number of such orbits, then the mass of this component must be smaller than two thirds of the total mass. The proofs use only the algebraic (i.e. not the number theoretic) properties of the toral automorphisms together with the exponential instability of the dynamics and therefore work in all dimensions.
The Index Of Sturmian Sequences
 European J. Combin
, 2000
"... We consider Sturmian sequences and provide an explicit formula for the index of such a sequence in terms of the continued fraction expansion coecients of its slope. ..."
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Cited by 15 (4 self)
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We consider Sturmian sequences and provide an explicit formula for the index of such a sequence in terms of the continued fraction expansion coecients of its slope.
Continued fractions and RSA with small secret exponent
 Tatra Mt. Math. Publ
"... Abstract. Extending the classical Legendre’s result, we describe all solutions of the inequality α − a/b  < c/b 2 in terms of convergents of continued fraction expansion of α. Namely, we show that a/b = (rpm+1 ±spm)/(rqm+1 ±sqm) for some nonnegative integers m, r, s such that rs < 2c. As an applic ..."
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Cited by 15 (6 self)
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Abstract. Extending the classical Legendre’s result, we describe all solutions of the inequality α − a/b  < c/b 2 in terms of convergents of continued fraction expansion of α. Namely, we show that a/b = (rpm+1 ±spm)/(rqm+1 ±sqm) for some nonnegative integers m, r, s such that rs < 2c. As an application of this result, we describe a modification of Verheul and van Tilborg variant of Wiener’s attack on RSA cryptosystem with small secret exponent. 1.
Digits and Continuants in Euclidean Algorithms. Ergodic versus Tauberian Theorems
, 2000
"... We obtain new results regarding the precise average case analysis of the main quantities that intervene in algorithms of a broad Euclidean type. We develop a general framework for the analysis of such algorithms, where the averagecase complexity of an algorithm is related to the analytic behaviou ..."
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Cited by 14 (5 self)
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We obtain new results regarding the precise average case analysis of the main quantities that intervene in algorithms of a broad Euclidean type. We develop a general framework for the analysis of such algorithms, where the averagecase complexity of an algorithm is related to the analytic behaviour in the complex plane of the set of elementary transformations determined by the algorithms. The methods rely on properties of transfer operators suitably adapted from dynamical systems theory and provide a unifying framework for the analysis of the main parameters digits and continuants that intervene in an entire class of gcdlike algorithms. We operate a general transfer from the continuous case (Continued Fraction Algorithms) to the discrete case (Euclidean Algorithms), where Ergodic Theorems are replaced by Tauberian Theorems.
Periodic boxcar deconvolution and Diophantine approximation
 Ann. of Statist
, 2004
"... We consider the nonparametric estimation of a periodic function that is observed in additive Gaussian white noise after convolution with a “boxcar,” the indicator function of an interval. This is an idealized model for the problem of recovery of noisy signals and images observed with “motion blur. ” ..."
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Cited by 14 (1 self)
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We consider the nonparametric estimation of a periodic function that is observed in additive Gaussian white noise after convolution with a “boxcar,” the indicator function of an interval. This is an idealized model for the problem of recovery of noisy signals and images observed with “motion blur. ” If the length of the boxcar is rational, then certain frequencies are irretreviably lost in the periodic model. We consider the rate of convergence of estimators when the length of the boxcar is irrational, using classical results on approximation of irrationals by continued fractions. A basic question of interest is whether the minimax rate of convergence is slower than for nonperiodic problems with 1/flike convolution filters. The answer turns out to depend on the type and smoothness of functions being estimated in a manner not seen with “homogeneous ” filters. 1. Introduction. 1.1. Statement of problem and motivation. Suppose that we observe Y(t) for t ∈[−1, 1],whereYisdrawn from an indirect estimation model in Gaussian white noise:
On the Classification of Rational Knots
 Enseign. Math
"... In this paper we give combinatorial proofs of the classification of unori ented and oriented rational knots based on the now known classifica tion of alternating knots and the calculus of continued fractions. We also characterize the class of strongly invertible rational links. Rational links ..."
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Cited by 12 (6 self)
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In this paper we give combinatorial proofs of the classification of unori ented and oriented rational knots based on the now known classifica tion of alternating knots and the calculus of continued fractions. We also characterize the class of strongly invertible rational links. Rational links are of fundamental importance in the study of DNA recombination.