Characterized subgroups of the circle group Characterized subgroups of the circle group Anna Giordano Bruno (University of Udine) 31st Summer Conference on Topology and its Applications Leicester - August 2nd, 2016 (Topological Methods in Algebra and Analysis) Characterized subgroups of the circle group Introduction p-torsion and torsion subgroup Introduction Let G be an abelian group and p a prime. The p-torsion subgroup of G is n tp(G) = fx 2 G : p x = 0 for some n 2 Ng: The torsion subgroup of G is t(G) = fx 2 G : nx = 0 for some n 2 N+g: The circle group is T = R=Z written additively (T; +); for r 2 R, we denoter ¯ = r + Z 2 T. We consider on T the quotient topology of the topology of R and µ is the (unique) Haar measure on T. 1 tp(T) = Z(p ). t(T) = Q=Z. Characterized subgroups of the circle group Introduction Topologically p-torsion and topologically torsion subgroup Let now G be a topological abelian group. [Bracconier 1948, Vilenkin 1945, Robertson 1967, Armacost 1981]: The topologically p-torsion subgroup of G is n tp(G) = fx 2 G : p x ! 0g: The topologically torsion subgroup of G is G! = fx 2 G : n!x ! 0g: Clearly, tp(G) ⊆ tp(G) and t(G) ⊆ G!. [Armacost 1981]: 1 tp(T) = tp(T) = Z(p ); e¯ 2 T!, bute ¯ 62 t(T) = Q=Z. Problem: describe T!. Characterized subgroups of the circle group Introduction Topologically p-torsion and topologically torsion subgroup [Borel 1991; Dikranjan-Prodanov-Stoyanov 1990; D-Di Santo 2004]: + For every x 2 [0; 1) there exists a unique (cn)n 2 NN such that 1 X cn x = ; (n + 1)! n=1 cn < n + 1 for every n 2 N+ and cn < n for infinitely many n 2 N+. Theorem (Dikranjan-Prodanov-Stoyanov 1990; D-Di Santo 2004) cn Let x 2 [0; 1). Then x¯ 2 T! , n+1 ! 0 in T: [Dikranjan-Di Santo 2004]: T! has Haar measure 0, T! has size c and T! is not divisible. Characterized subgroups of the circle group Topologically u-torsion subgroup Definition Topologically u-torsion subgroup Definition (Dikranjan-Prodanov-Stoyanov 1990; Dikranjan 2001) Let G be a topological abelian group and u = (un)n 2 ZN. The topologically u-torsion subgroup of G is tu(G) = fx 2 G : unx ! 0g: n n tp(G) = fx 2 G : p x ! 0g = tp(G), where p = (p )n. G! = fx 2 G : n!x ! 0g = tu(G), where u = ((n + 1)!)n. Example (Dikranjan-Kunen 2007) t(T) = Q=Z = tu(T), where u is the sequence (1!; 2!; 2·2!; 3!; 2·3!; 3·3!; 4!;:::; n!; 2·n!; 3·n!;:::; n·n!; (n+1)!;:::). Problem: given u 2 ZN, describe tu(T). Characterized subgroups of the circle group Topologically u-torsion subgroup Arithmetic sequences n Which property is shared by (p )n and ((n + 1)!)n? Let u = (un)n be an arithmetic sequence (i.e., u0 = 1, un 2 N+ and un j un+1 for every n 2 N) and for every n 2 N let u un+1 dn+1 = 2 N+: un Theorem u N+ For every x 2 [0; 1), there exists a unique (cn (x))n 2 N with 1 X cu(x) x = n ; u n=1 n u u cn (x) < dn for every n 2 N+, u u and cn (x) < dn − 1 for infinitely many n 2 N+. Characterized subgroups of the circle group Topologically u-torsion subgroup Arithmetic sequences Theorem (Dikranjan-Prodanov-Stoyanov 1990; D-Di Santo 2004) u Let x = P1 cn (x) 2 [0; 1). n=1 un u u If (dn )n is bounded, then x¯ 2 tu(T) , (cn (x))n is ev. 0: u u cn (x) If d ! +1, then x¯ 2 tu( ) , u ! 0 in : n T dn T [Dikranjan-Di Santo 2004, Dikranjan-Impieri 2014]: Complete description of tu(T) for u arithmetic sequence. Corollary The following conditions are equivalent: u (dn )n is bounded; tu(T) is countable; tu(T) is torsion. Characterized subgroups of the circle group Characterized subgroups of T Definition and generalization Characterized subgroups of T Definition (B´ır´o-Deshouillers-S´os2001) A subgroup H of T is characterized if H = tu(T) for some u 2 ZN. (H is characterized by u; u characterizes H.) Problem Describe the characterized subgroups of T. In other words, given u 2 ZN, describe tu(T). The \inverse problem": given H ≤ T, is H characterized? Problem (Di Santo 2002; Maharam-Stone 2001) Given H ≤ T characterized, describe SH = fu 2 ZN : H = tu(T)g ≤ ZN. Characterized subgroups of the circle group Characterized subgroups of T First properties and results If H is a finite subgroup of T, then H is characterized. u characterizes T if and only if u is eventually zero. If H T is characterized, then: N H = tu(T) for u 2 N+ strictly increasing; µ(H) = 0. Since t ( ) = T S T x 2 : ku xk ≤ 1 u T N≥2 m2N n≥m T n N is a Borel set, so either tu(T) is countable or jtu(T)j = c: Theorem (Eggleston 1952) un+1=un ! +1 ) jtu(T)j = c. (un+1=un)n bounded ) tu(T) countable. Characterized subgroups of the circle group Characterized subgroups of T First properties and results Theorem (Borel 1983) All countable subgroups of T are characterized by some u 2 ZN. [Beiglb¨ock-Steineder-Winkler2006]: u can be chosen with (un+1=un)n bounded, but also arbitrarily fast increasing. Borel's motivation: (xn)n 2 RN is uniformly distributed mod 1 if for all [a; b] ⊆ [0; 1), jfj 2 f0;:::; ng : fx g 2 [a; b]gj j −! a − b: n Theorem (Weyl, 1916; Kuipers-Niederreiter 1974) Let u 2 NN be a strictly increasing sequence. Then λ(fβ 2 R :(funβg)n is uniformly distributed mod 1g) = 1. This does not hold for all β 2 R, so study the \opposite case" ¯ ¯ tu(T) = fβ 2 T : unβ ! 0g. Characterized subgroups of the circle group Characterization of the cyclic subgroups of T Continued fractions The infinite cyclic subgroups of T Let α be an irrational number and consider hα¯i ≤ T. The irrational α has a unique continued fraction expansion 1 α = a0 + 1 a1 + a2 + ::: denoted by α = [a0; a1; a2;:::]. N For every n 2 N, let q = (qn)n 2 N+, where pn [a0; a1;:::; an] = ; with qn > 0: qn Since qnα¯ ! 0 in T, so hα¯i ⊆ tq(T): Characterized subgroups of the circle group Characterization of the cyclic subgroups of T Larcher Theorem Theorem (Larcher 1988) hα¯i = tq(T) when (an)n is bounded. The equality does not hold true in general, more precisely: Theorem (Kraaikamp-Liardet 1992) hα¯i = tq(T) if and only if (an)n is bounded. Example p 1+ 5 Consider the Golden ratio ' = 2 : Then h'¯i = tf (T); where f = (fn)n is the Fibonacci sequence f0 = 1; f1 = 1; f2 = 2; f3 = 3; f4 = 5;:::: Indeed, ' = [1; 1; 1;:::] and qn = fn for every n 2 N+. Characterized subgroups of the circle group Characterization of the cyclic subgroups of T A generalization: recurrent sequences [Barbieri-Dikranjan-Milan-Weber 2008]: Sequences u 2 ZN verifying a linear recurrence. For every n ≥ 2, un = anun−1 + bnun−2; an; bn 2 N+: Theorem (Barbieri-Dikranjan-Milan-Weber 2008) jtu(T))j = c , (un+1=un)n not bounded: If α = [a0; a1;:::] is an irrational and q = (qn)n, then for every n ≥ 2, qn = anqn−1 + qn−2; q0 = 1; q1 = a1: Corollary The following conditions are equivalent: hα¯i = tq(T); (an)n is bounded; tq(T) is countable. Characterized subgroups of the circle group Characterization of the cyclic subgroups of T B´ır´o-Deshouillers-S´osTheorem Let α = [a0; a1;::: ] irrational, q = (qn)n as above and let vα: q0≤q1<2q1 < : : : < a2q1 < q2<2q2 < : : : < a3q2 < q3<2q3 < : : : Theorem (B´ır´o-Deshouillers-S´os2001) hα¯i = tvα (T). Problem (B´ır´o-Deshouillers-S´os2001) Find all sequences u 2 ZN characterizing hα¯i. In particular, if q ⊆ u ⊆ vα, one has hα¯i = tvα (T) ⊆ tu(T) ⊆ tq(T); and the question is: under which hypotheses one has hα¯i = tu(T)? Theorem (Marconato 2016) If (un+1=un)n is bounded, then hα¯i = tvα (T) = tu(T). Characterized subgroups of the circle group Characterized subgroups of T and precompact group topologies of Z TB-sequences in Z Precompact topologies on Z with converging sequences [Raczkowski 2002; Barbieri-Dikranjan-Milan-Weber 2003]: τ Given u 2 ZN, 9 a precompact gr. top. τ on Z such that un −! 0? A topological abelian group (G; τ) is: totally bounded if 8 ; 6= U 2 τ, 9F ⊆ G finite, G = U + F ; precompact if τ is Hausdorff and totally bounded. For G abelian group and H ≤ Gb, let TH be the weakest group topology on G such that all χ 2 H are continuous. Theorem (Comfort-Ross 1964) TH is totally bounded and w(G; TH ) = jHj. If (G; τ) is totally bounded, then τ = TH for some H ≤ Gb. TH is Hausdorff if and only if H is dense in Gb. For G = Z, we have Gb = T. Characterized subgroups of the circle group Characterized subgroups of T and precompact group topologies of Z TB-sequences in Z Let u 2 ZN and τ a totally bounded group topology on Z.