Rationally almost periodic sequences, polynomial multiple recurrence and symbolic dynamics

Abstract

A set R ⊂ N is called rational if it is well-approximable by finite unions of arithmetic progressions, meaning that for every ǫ > 0 there exists a set B = r i=1 a i N+b i , where a 1 , . . . , a r , b 1 , . . . , b r ∈ N, such thatExamples of rational sets include many classical sets of number-theoretical origin such as the set of squarefree numbers, the set of abundant numbers, or sets of the form Φ x := {n ∈ N :< x}, where x ∈ [0, 1] and ϕ is Euler's totient function. We investigate the combinatorial and dynamical properties of rational sets and obtain new results in ergodic Ramsey theory. Among other things, we show that if R ⊂ N is a rational set with d(R) > 0, then the following are equivalent:(b) R is an averaging set of polynomial single recurrence.(c) R is an averaging set of polynomial multiple recurrence.As an application, we show that if R ⊂ N is rational and divisible, then for any set E ⊂ N with d(E) > 0 and any polynomials p i ∈ Q[t], i = 1, . . . , ℓ, which satisfy p i (Z) ⊂ Z and p i (0) = 0 for all i ∈ {1, . . . , ℓ}, there exists β > 0 such that the sethas positive lower density.Ramsey-theoretical applications naturally lead to problems in symbolic dynamics, which involve rationally almost periodic sequences (sequences whose level-sets are rational). We prove that if A is a finite alphabet, η ∈ A N is rationally almost periodic, S denotes the left-shift on A Z and X := {y ∈ A Z : each finite word appearing in y appears in η}, then η is a generic point for an S-invariant probability measure ν on X such that the measure preserving system (X, ν, S) is ergodic and has rational discrete spectrum.

Sarnak’s Conjecture: What’s New

“…69 While in [53,60] Möbius disjointness is proved for the dynamical systems given by bijective substitutions, [48] treats the opposite case, so called synchronized. As noted in [21], this leads to dynamical systems given by rational sequences and such are Möbius disjoint. Note also that for the synchronized case, once the system is uniquely ergodic, it is automatically a uniquely ergodic model of an automorphism with discrete spectrum, cf.…”

“…For general B-free sets the situation is more complicated and we have the following result: This can be generalized to B that are not Besicovitch by considering divisibility and recurrence along a certain subsequence pN k q kě1 . As a combinatorial application, one obtains in [21] the following result: Suppose that pN k q kě1 is such that the density of F B along pN k q kě1 exists and is positive. Then there exists D Ă F B which equals F B up to a set of zero density along pN k q kě1 such that for all r P D and for all E Ă N with positive upper density, for any polynomials p i P Qrts, i " 1, .…”

“…Clearly, this leads us to study the behavior of different (prime) powers of a fixed map T . We should warn the reader that when applying Theorem 2.13, we do not expect to have (21) satisfied for all continuous functions, in fact, even in uniquely ergodic systems, in general, it cannot hold for all zero mean functions 21 but we need a subset of CpXq which is linearly dense, cf. footnote 4.…”

“…[21]). Given B Ă N that is Besicovitch, there exists a set D Ă F B with dpF B zDq " 0 such that the set F B´r is an averaging set of polynomial multiple recurrence if and only if r P D. Moreover, D " F B if and only if the set B is taut.…”

A Structure Theorem for Level Sets of Multiplicative Functions and Applications

“…(2) χ(n) = 0 whenever gcd(d, n) > 1, and χ(n) is a ϕ(d)-th root of unity whenever gcd(d, n) = 1, where ϕ denotes Euler's totient function; (3) χ(nm) = χ(n)χ(m) for all n, m ∈ N. Any Dirichlet character is periodic and completely multiplicative. 3 We also remark that χ : N → C is a Dirichlet character of modulus k if and only if there exists a group character χ of the multiplicative group (Z/kZ) * such that χ(n) = χ(n mod k) for all n ∈ N. The Dirichlet character determined by the trivial (constant equal to 1) character of (Z/kZ) * is called the principal character of modulus k. It is denoted by χ 1 . Note that if d|k and χ is a Dirichlet character of modulus d then…”