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# hw4-ERG - DynSys MTG 6401 Ergodic HW tssera e Prof JLF King...

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DynSys MTG 6401 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. ( I’ll connect it to Ergodic Theory in the Addendum . ) Let Δ( z ) [ 1 .. 9 ] denote the high-order digit of the base-ten numeral for z R . So Δ( π / 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 Δ(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 + | Δ( n ) = d } . In HW3, I sketched an argument showing that L 1 = 1 9 and U 1 = 5 9 . Please compute the other densities. Which values of d make the ratio U d /L d an integer? Addendum . In HW1, you showed that the × 2 doubling map on ( R + , Leb) gave actual densities ( i.e, UpperDen=LowerDen ) to sets { E d } 9 d =1 .
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