Resolution of T. Ward’s Question and the Israel–Finch Conjecture. Precise Asymptotic Analysis of an Integer Sequence Motivated by the Dynamical Mertens’ Theorem for Quasihyperbolic Toral Automorphisms Mark Daniel Ward, ∗ Department of Statistics, Purdue University May 31, 2013 Joint work with Jeffrey B. Gaither and Guy Louchard Mark Daniel Ward (Purdue University) Resolution of the Israel–Finch Conjecture May 31, 2013 1 / 48
� 1 � n j =1 4 sin 2 ( π jx ) dx Origins of the Problem a n := 0 Peter Walters introduced this problem, in the Transactions of the American Mathematical Society , 140:95–107, 1969: Mark Daniel Ward (Purdue University) Resolution of the Israel–Finch Conjecture May 31, 2013 2 / 48
� 1 � n j =1 4 sin 2 ( π jx ) dx Origins of the Problem a n := 0 Thomas Ward introduced this sequence on OEIS on 7 Jan 2008, stating: newsgroup sci.math.research , from Thomas Ward on 16 Jan 2008: Mark Daniel Ward (Purdue University) Resolution of the Israel–Finch Conjecture May 31, 2013 3 / 48
� 1 � n j =1 4 sin 2 ( π jx ) dx Origins of the Problem a n := 0 After getting our solution, Thomas Ward sent us a great number of emails, mentioning other related problems. These might be of interest to people interested in Dynamics, or Ergodic Theory, with applications to Number Theory. For instance: ◮ “Realizable sequences: a non-trivial property of an integer sequence is to be the count of points of period n for some map—this is a congruence and a positivity condition.” ◮ “Products and iterates”, maps, enumeration of orbits: “this generates a lot of ‘nice’ sequences, some well-known, some mysterious. In (very) simple examples we have used Perron to find asymptotics, but I think there is rich territory here for cleverer things.” Mark Daniel Ward (Purdue University) Resolution of the Israel–Finch Conjecture May 31, 2013 4 / 48
� 1 � n j =1 4 sin 2 ( π jx ) dx Origins of the Problem a n := 0 ◮ “Some notorious problems. . . that are very easy to analyze in one dimension that become very problematical in two.” . . . “Let a ( n ) be the number of ways to fill an n × n square with 0s and 1s so that you never have two 1s in a row vertically nor horizontally. Easy to argue that a ( n ) ∼ C n for some C ∈ (1 , 2) (and there are some estimates) but we have no clue about the real picture. This is probably extremely difficult, and there is no a priori thing that makes you expect a ‘closed form’ asymptotic.” Mark Daniel Ward (Purdue University) Resolution of the Israel–Finch Conjecture May 31, 2013 5 / 48
� 1 � n j =1 4 sin 2 ( π jx ) dx Origins of the Problem a n := 0 ◮ “Moonshine: one-dimensional subshifts of finite type give rise in a natural way to sequences with rational associated zeta functions with growth rates parameterised by a certain nice countable collection of numbers, the Perron numbers.” ◮ “Twice-distilled moonshine: there is a natural analytic toolbox associated to certain classes of problems: ◮ ’hyperbolic’ dynamics—zeta functions with meromorphic extensions beyond radius of convergence - Tauberian methods ◮ Slow growth dynamics - orbit Dirichlet series - elementary methods (analytically too nasty for Tauberian methods)” Mark Daniel Ward (Purdue University) Resolution of the Israel–Finch Conjecture May 31, 2013 6 / 48
� 1 � n j =1 4 sin 2 ( π jx ) dx Origins of the Problem a n := 0 From a topological/dynamical perspective: a n is the asymptotic coefficient in the weighted sum of the orbital numbers of a certain toral automorphism. The study of this automorphism is motivated by the search for a topological analogue to Mertens’ prime number theorem . There is a strong structural similarity between the distribution of the prime integers and that of the orbits of an automorphism acting on an n-torus T n = S 1 × · · · × S 1 . Mark Daniel Ward (Purdue University) Resolution of the Israel–Finch Conjecture May 31, 2013 7 / 48
� 1 � n j =1 4 sin 2 ( π jx ) dx Origins of the Problem a n := 0 Specifically, the classical Mertens’ Theorem gives 1 � � � ∼ log( N ) , 1 − 1 / p p ≤ N and the analogous hyperbolic toral diffeomorphism result is 1 � M T ( N ) := e h | τ | ∼ log( N ) , (1) | τ |≤ N where each τ = { x , T ( x ) , . . . , T k ( x ) = x } denotes a closed orbit of length k , and h represents the topological entropy. Mark Daniel Ward (Purdue University) Resolution of the Israel–Finch Conjecture May 31, 2013 8 / 48
� 1 � n j =1 4 sin 2 ( π jx ) dx Origins of the Problem a n := 0 Noorani (1999) showed that when T is merely an ergodic toral automorphism (such an automorphism is said to quasihyperbolic ), one has M T ( N ) = m log( N ) + C 1 + o (1) (2) for some positive integer m , where a n is precisely the m in (2) for a specific automorphism (which we will describe in a moment). Mark Daniel Ward (Purdue University) Resolution of the Israel–Finch Conjecture May 31, 2013 9 / 48
� 1 � n j =1 4 sin 2 ( π jx ) dx Origins of the Problem a n := 0 Jaidee, Stevens, and Ward (2011) improved Noorani’s estimate and also showed that the constant m is given by n � � 4 sin 2 ( π x j ) d x 1 · · · d x j , m = (3) X j =1 where X is found as follows: Write e ± 2 π i θ 1 , . . . , e ± 2 π i θ t as the eigenvalues of modulus 1 of the matrix A that defines the automorphism T , and let X ⊂ T d denote the closure of the set { ( k θ 1 , . . . , k θ t ) : k ∈ Z } in T d . Mark Daniel Ward (Purdue University) Resolution of the Israel–Finch Conjecture May 31, 2013 10 / 48
� 1 � n j =1 4 sin 2 ( π jx ) dx Origins of the Problem a n := 0 The particular toral automorphism that gives rise to T. Ward’s a n has been of interest since Walter (1969), who introduced the automorphism given by 0 0 0 − 1 1 0 0 8 A = 0 1 0 − 6 0 0 1 8 on the 4-torus T 4 = S 1 × S 1 × S 1 × S 1 for an example of an affine mapping between compact connected metric abelian groups, for which the mapping commutes only with continuous maps that are also affine. For this matrix A , 2 of the 4 eigenvalues have modulus 1, namely: √ √ � 2 − 3 ± i 4 3 − 6 . √ � � 1 2 − 3 √ So the θ 1 from the previous slide is θ 1 = 2 π arctan . √ − 6+4 3 Mark Daniel Ward (Purdue University) Resolution of the Israel–Finch Conjecture May 31, 2013 11 / 48
� 1 � n j =1 4 sin 2 ( π jx ) dx Origins of the Problem a n := 0 Noorani (1999) further cited A as an example of a strictly quasi-hyperbolic automorphism on the torus. Jaidee et al. later considered A as their example of a toral automorphism whose asymptotic coefficient m = 6 exceeded 2 t = 2 (where t = 1 since A has 2 = 2 t eigenvalues on the unit circle). They also used A to define the automorphism A 1 ⊕ A 2 ⊕ · · · ⊕ A n on the 4 n -torus T 4 n . This defines the context: The a n ’s are precisely the coefficient m of log( N ) in the asymptotic growth of M T ( n ) (from slide 8), with T 4 n the torus under consideration and A 1 ⊕ · · · ⊕ A n the quasihyperbolic automorphism. Mark Daniel Ward (Purdue University) Resolution of the Israel–Finch Conjecture May 31, 2013 12 / 48
� 1 � n j =1 4 sin 2 ( π jx ) dx Origins of the Problem a n := 0 So we analyze the precise first-order asymptotics of ( a n ) n ≥ 1 = (2 , 4 , 6 , 10 , 12 , 20 , 24 , 34 , 44 , 64 , . . . ) . A133871 from the On-Line Encyclopedia of Integer Sequences). Mark Daniel Ward (Purdue University) Resolution of the Israel–Finch Conjecture May 31, 2013 13 / 48
� 1 � n j =1 4 sin 2 ( π jx ) dx Israel’s Conjecture for a n := 0 newsgroup sci.math.research , from Robert Israel on 17 Jan 2008: Conjecture about the logarithm of a n : ln( a n 2 − n ) → − 0 . 3 . . . , or equivalently ln a n → (ln (2) − 0 . 3 . . . ) n . n Mark Daniel Ward (Purdue University) Resolution of the Israel–Finch Conjecture May 31, 2013 14 / 48
� 1 � n j =1 4 sin 2 ( π jx ) dx Finch’s Conjecture for a n := 0 http://www.people.fas.harvard.edu/~sfinch/ many newer “Supplementary Materials,” including: Mark Daniel Ward (Purdue University) Resolution of the Israel–Finch Conjecture May 31, 2013 15 / 48
� 1 � n j =1 4 sin 2 ( π jx ) dx Finch’s Conjecture for a n := 0 Mark Daniel Ward (Purdue University) Resolution of the Israel–Finch Conjecture May 31, 2013 16 / 48
� 1 � n j =1 4 sin 2 ( π jx ) dx Finch’s Conjecture for a n := 0 Finch wrote a n in a very simple way : Finch’s conjecture about the logarithm of a n was: a 1 / n ∼ 1 . 48 . . . ∼ 2 e − 0 . 29 ... . n This conjecture is equivalent to Israel’s conjecture. Finch has several other conjectures in this essay. He says: Mark Daniel Ward (Purdue University) Resolution of the Israel–Finch Conjecture May 31, 2013 17 / 48
� 1 � n j =1 4 sin 2 ( π jx ) dx Representations of a n := 0 � 1 n � (1 − e 2 π ijx )(1 − e − 2 π ijx ) dx Equivalently, a n := 0 j =1 � 1 � 1 e 2 π i ℓ x dx = 0 for ℓ ∈ Z \{ 0 } , e 2 π i 0 x dx = 1 . Note: and 0 0 So we expand � n j =1 (1 − e 2 π ijx )(1 − e − 2 π ijx ) and then integrate: only the terms in which the j’s sum to 0 will play a nontrivial role. If the integral of a term is nonzero, it is either 1 or − 1, depending on whether an even or odd number of j ’s were involved in the product. Mark Daniel Ward (Purdue University) Resolution of the Israel–Finch Conjecture May 31, 2013 18 / 48
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