Lecture1

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Unformatted text preview: Lecture 1 : Introduction We will start with a simple combinatorial problem. Consider {- 1 , 1 } 1000 . How many elements x ∈ {- 1 , 1 } 1000 satisfy fl fl fl fl 1000 X i =1 x i fl fl fl fl ≥ 50? More generally, for any n ∈ N and λ > 0 how many elements x ∈ {- 1 , 1 } n satisfy fl fl fl fl n X i =1 x i fl fl fl fl ≥ λn ? The answer is given by the binomial distribution. We are only seeking approximations. This is a question that we will spend a fair deal of time on this quarter. Today we will be satisfied with a crude upper bound. Fact: For any r ∈ R ( r + 1) 2 + ( r- 1) 2 = ( r 2 + 2 r + 1) + ( r 2- 2 r + 1) = 2( r 2 + 1) . For x ∈ {- 1 , 1 } n we write S n ( x ) ∑ n i =1 x i and for m < n we write x | m for the restriction of x to the first m terms. Lemma 1.0.1 X x ∈{- 1 , 1 } n ( S n ( x )) 2 = n 2 n . Proof: By induction. It is easy to check that it is true for n = 1. Assume it is true for n ....
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