hw4-ERG - DynSys MTG6401 Ergodic HW t essera Prof. JLF King...

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Unformatted text preview: DynSys MTG6401 Ergodic HW t essera Prof. JLF King 5Nov2009 ( Due Wednesday, 07Oct. Please staple this sheet as the first page of your write-up. ) Prolegomenon. To give you guys a break, here is a undergraduate combinatorics problem. ( Ill connect it to Ergodic Theory in the Addendum . ) Let ( z ) [ 1 .. 9 ] denote the high-order digit of the base-ten numeral for z R . So ( / 100) is 3. ( Oh, we have a measure-zero set to fret about. For dyadic rationals z 6 =0, use the decimal expansion which is eventually the digits 000 forevermore. As for z =0, de- fine (0) as you see fit. ) Below, let log() denote log 10 (), and let K be the unit circle. H8: For each digit d { 1 , 2 ,..., 9 } let U d and L d be the upper and lower densities of the set E d := { n Z + | ( n ) = d } . In HW3, I sketched an argument showing that L 1 = 1 9 and U 1 = 5 9 . Please compute the other densities....
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This note was uploaded on 01/26/2012 for the course MTG 6401 taught by Professor Staff during the Fall '09 term at University of Florida.

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