EXERCISES FOR GEOMETRY OF NUMBERS WEEK 8: 01.05.2020 Exercise 1. 1. Consider all the intervals of the form [a2t; (a + 1)2t] with a; t non- ∗ negative integers. For s 2 N , let Ls be the collection of such intervals with t ≤ s − 1 s s contained in [0; 2 ]. Show that for any k 2 N with k ≤ 2 , the interval [0; k] can be covered by at most s elements of Ls. 2. Let F : [0; 1] × (0; 1) be a bounded measurable function and suppose that there exists c > 0 such that for every interval (a; b) ⊂ (0; 1), we have Z 1 Z b 2 F (x; t)dt dx ≤ c(b − a): 0 a Show that 2 X Z 1 Z F (x; t)dt dx ≤ cs2s: 0 I I2Ls ∗ 3. Fix > 0. Show that for every s 2 N , there exists a measurable set As ⊂ [0; 1] such −1−" 1 s that jAsj = O(s ) and for every y2 = As and for every k 2 N with k ≤ 2 , we have Z k s 3 + j F (t; y)dtj ≤ 2 2 s 2 : 0 4. Deduce from 3. 2 that for a.e. y 2 [0; 1], we have Z T 1 − 1 3 j F (y; t)dtj = O(T 2 log(T ) 2 ) T 0 Although the arithmetic functions are not the main focus of the course, the next two exercises touch upon those and aim at proving the arithmetic estimate used in Schmidt's a.s. counting theorem. Exercise 2 (Moebius inversion). A natural number n 2 N is called square-free if no ∗ square of a prime divides n. Recall that the Moebius function µ : N ! C is defined as µ(n) = 0 if n is not square-free, µ(n) = 1 if n has an even number of prime divisors and µ(n) = −1 if n has an odd number of prime divisors. 1. Show that µ is multiplicative (i.e. if d1; d2 are relatively prime then µ(d1d2) = µ(d1)µ(d2)) and X X µ(d) = 1 if n = 1 and µ(d) = 0 if n > 1 djn djn ∗ 2. On the set of arithmetic functions (i.e. functions N ! C), define the operation ∗ by setting X (f ∗ g)(n) = f(d)g(n=d): djn 1 Hint: see the construction of the set Bs in the proof of Schmidt's theorem in lecture notes 2As usual, if you cannot show what is required in step n of the exercise, you are encouraged to use the result of n in the consideration of the step n + k, k ≥ 1. 1 2 EXERCISES FOR GEOMETRY OF NUMBERS WEEK 8: 01.05.2020 Show that ∗ is a commutative and associative operation. 3. Let be the arithmetic function defined by (1) = 1 and (n) = 0 for every n > 1. Show that it is an identity element, i.e. ∗ f = f for every arithmetic function f. P 4. Using 1., 2. and 3. deduce the Moebius inversion formula: if g(n) = djn f(d), then P f(n) = djn µ(d)g(n=d). Exercise 3 (Proving the arithmetic estimate used in Schmidt's a.s. counting theorem). All symbols in this exercise stand for positive integers. Recall that we denote by '(k; q) the number of n ≤ q such that (n; q) ≤ k, where (n; q) stands for the greatest common divisor. 1. Show that X q '(k; q) = '( ): w w≤k wjq P 3 2. Show that n = djn '(d) where ' is Euler's totient function. P −1 3. Using 2. and Moebius inversion formula, show that '(x) = x yjx µ(y)y . 4. Deduce that b v c v minfk;vg bv=wc yw X X X X '(k; q)q−1 = w−1 µ(y)y−1 1: q=1 w=1 y=1 q=1 5. Deduce 4 v minfk;vg bv=wc X X X '(k; q)q−1 = v w−2 µ(y)y−2 + O(log v log k): (1) q=1 w=1 y=1 P −2 −1 6. Using the fact that y≥1 µ(y)y = ζ(2) , conclude that (1) is equal to minfk;vg minfk;vg X X v w−2ζ(2)−1 + O( w−1) + O(log v log k) w=1 w=1 which in turn is equal to v + O(vk−1 + log v log k): 3Hint: it suffices to treat the prime numbers, and prove that ' is multiplicative in the same sense as in the previous exercise. To show multiplicativity, one can use Chinese remainder theorem or prove that if r1 rs Qs 1 n = p : : : ps , then '(n) = n (n − ) 1 k=1 pk 4 Px 1 Recall that y=1 y = O(log x).