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Descriptive set theory of families of small sets
 J. Symbolic Logic
"... Abstract. This is a survey paper on the descriptive set theory of hereditary families of closed sets in Polish spaces. Most of the paper is devoted to ideals and σideals of closed or compact sets. ..."
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Abstract. This is a survey paper on the descriptive set theory of hereditary families of closed sets in Polish spaces. Most of the paper is devoted to ideals and σideals of closed or compact sets.
Trichotomies for ideals of compact sets
 J. SYMBOLIC LOGIC
"... We prove several trichotomy results for ideals of compact sets. Typically, we show that a “sufficiently rich” universally Baire ideal is either Π 0 3hard, or Σ 0 3hard, or else a σideal. ..."
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We prove several trichotomy results for ideals of compact sets. Typically, we show that a “sufficiently rich” universally Baire ideal is either Π 0 3hard, or Σ 0 3hard, or else a σideal.
Mixing operators and small subsets of the circle
, 2011
"... Abstract. We provide complete characterizations, on Banach spaces with cotype 2, of those linear operators which happen to be weakly mixing or strongly mixing transformations with respect to some nondegenerate Gaussian measure. These characterizations involve two families of small subsets of the ci ..."
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Abstract. We provide complete characterizations, on Banach spaces with cotype 2, of those linear operators which happen to be weakly mixing or strongly mixing transformations with respect to some nondegenerate Gaussian measure. These characterizations involve two families of small subsets of the circle: the countable sets, and the socalled sets of uniqueness for FourierStieltjes series. The most interesting part, i.e. the sufficient conditions for weak and strong mixing, is valid on an arbitrary (complex, separable) Fréchet space. 1.
VARIATIONS ON A THEME OF DEBS AND SAINT
"... Abstract. A famous theorem of Debs and Saint Raymond states that the complement of a set of first category is of strong multiplicity. We prove a theorem which combines this with a result of Rudin which states that independent closed sets of strong multiplicity exist. We also prove a theorem which co ..."
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Abstract. A famous theorem of Debs and Saint Raymond states that the complement of a set of first category is of strong multiplicity. We prove a theorem which combines this with a result of Rudin which states that independent closed sets of strong multiplicity exist. We also prove a theorem which combines the theorem of Debs and Saint Raymond with a theorem of Wiener and Wintner which states that there exists a measure with singular support whose convolution square is absolutely continuous. 1.
ON SOME ERRORS RELATED TO THE GRADUATION OF MEASURING INSTRUMENTS
, 2006
"... Abstract. The error on a real quantity Y due to the graduation of the measuring instrument may be approximately represented, when the graduation is regular and fines down, by a Dirichlet form on R whose square field operator does not depend on the probability law of Y as soon as this law possesses a ..."
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Abstract. The error on a real quantity Y due to the graduation of the measuring instrument may be approximately represented, when the graduation is regular and fines down, by a Dirichlet form on R whose square field operator does not depend on the probability law of Y as soon as this law possesses a continuous density. This feature is related to the “arbitrary functions principle ” (Poincaré, Hopf). We give extensions of this property to R d and to the Wiener space for some approximations of the Brownian motion. We use a Girsanov theorem for Dirichlet forms which has its own interest. Connections are given with the discretization of stochastic differential equations.
Sets of extended uniqueness and σporosity
 COMMENT.MATH.UNIV.CAROLIN. 38,2 (1997)337–341
, 1997
"... We show that there exists a closed nonσporous set of extended uniqueness. We also give a new proof of Lyons ’ theorem, which shows that the class of H(n)sets is not large in U0. ..."
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We show that there exists a closed nonσporous set of extended uniqueness. We also give a new proof of Lyons ’ theorem, which shows that the class of H(n)sets is not large in U0.