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Baire - Baire Category Probabilistic Constructions and...

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Baire Category, Probabilistic Constructions and Convolution Squares T. W. K¨orner August 13, 2009 Contents 1 Introduction 2 2 Baire’s theorem 3 3 The Hausdorff metric 6 4 Independence and Kronecker sets 10 5 Besicovitch Sets 14 6 Measures 18 7 A theorem of Rudin 20 8 The poor man’s central limit theorem 24 9 Completion of the construction 28 10 Sets of uniqueness and multiplicity 35 11 Distributions 42 12 Debs and Saint-Reymond 43 13 The perturbation argument 47 14 Convolution of distinct measures 51 15 The Wiener–Wintner theorem 54 1
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16 Hausdorff dimension and measures 58 17 Thick Wiener–Wintner measures 67 18 More probability 70 19 Point masses to smooth functions 77 20 Hausdorff dimension and sums 86 21 The final construction 89 22 Remarks 94 1 Introduction In the past few years I have written a number of papers using simple Baire category and probabilistic results. The object of this course is to give exam- ples of the main theorems and the methods used to obtain them. Although we shall obtain other results, our main concern will be the question. Knowing something about the measure μ , what can we say about its convolution with itself (that is to say, the convolution square) μ μ ? This question goes back at least as far as the paper of Wiener and Wintner [28] in which they show that the convolution square of a singular measure need not be singular. Those who already know about such things may find it useful to see some of our main results. Those who do not, should be reassured that we will provide appropriate definitions and background in due course. We give a new proof of the following theorem of Besicovitch. Theorem 5.4. There exists a closed bounded set of Lebesgue measure con- taining lines of length at least 1 in every direction. We prove a quantitive verision of a theorem of Rudin. Theorem 7.3. Suppose that φ : N R is a sequence of strictly positive numbers with r α φ ( r ) → ∞ as r → ∞ whenever α> 0 . Then there exists a probability measure μ such that φ ( | r | ) ≥ | ˆ μ ( r ) | for all r negationslash = 0 , but supp μ is independent. We prove an extension (found independently by Matheron and Zelen´y) of a celebrated theorem of Debs and Saint Reymond. Theorem 12.5. Let B be a set of first category in T . Then we can find a probability measure μ with ˆ μ ( r ) 0 as | r | → ∞ such that supp μ is 2
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independent and the subgroup G of T generated by supp μ satisfies G B ⊆ { 0 } . We produce two substantial extensions of the theorem of Wiener and Wintner. The first is related to the theorem of Debs and Saint Reymond. Theorem 15.1. Let A be a set of first category in T . Then we can find a probability measure μ such that supp μ A = but d ( μ μ ) t = f ( t ) dt where f is a Lebesgue L 1 function. The second further extends a result of Saeki.
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