Hewitt-Marczewski-Pondiczery type theorem for abelian groups and Markovs potential density

HEWITT-MARCZEWSKI-PONDICZER Y TYPE THEOREM F OR ABELIAN GR OUPS AND MARK O V’S POTENT I AL DENSIT Y DIKRAN DIKRAN JAN AND DMITRI SHAKHMA T OV Abstra ct. F or an uncountable cardinal τ and a su bset S of an ab elian group G , the fol lo wing conditions are equiv alen t: (i) |{ ns : s ∈ S }| ≥ τ for all integers n ≥ 1; (ii) t here ex ists a group homomorphism π : G → T 2 τ such that π ( S ) is den se in T 2 τ . Moreo ver , if | G | ≤ 2 2 τ , then the follow ing item can be added to this list: (iii) th ere exists an isomorphism π : G → G ′ b etw een G and a subgroup G ′ of T 2 τ such t hat π ( S ) is dense in T 2 τ . W e p ro ve that the follo wi ng conditions are equiv alent for an uncountable subset S of an ab elian group G that is either (almost) torsion-free or divisible: (a) S is T -dense in G for some Hausdorff group topology T on G ; (b) S is T -dense in some precompact H ausdorff group top ology T on G ; (c) |{ ns : s ∈ S }| ≥ min  τ : | G | ≤ 2 2 τ  for ever y integer n ≥ 1. This partially resolves a qu estion of Mark ov going bac k t o 19 46. W e use N and P to denote the set of p ositiv e natural num b ers and prime num b ers, resp ectiv ely . The symb ol c denotes th e cardinality of the con tin uum, a nd ω denotes the first infi nite cardinal. F or an infinite cardinal κ , let log κ = min { τ ≥ ω : κ ≤ 2 τ } . The group of in teger num b ers is den oted b y Z . F or a su bset X of an abelian group G , w e u se h X i to denote the sub group of G generate d by X , and let nX = { nx : x ∈ X } for ev ery n ∈ Z . As usual, t ( G ) = { g ∈ G : ng = 0 f or some n ∈ N } denotes the tor sion p art of G . An ab elian group G is said to b e: (i) torsion -fr e e if t ( G ) = { 0 } ; (ii) torsio n if t ( G ) = G ; (iii) b ounde d if nG = { 0 } for some n ∈ N ; (iv) divisible if nG = G for ev ery n ∈ N . Recall that a group homomorphism ϕ : G → H from a n ab elian group G to an ab elian group H is a monomorphism if ker ϕ = { 0 } . A topological group is pr e c omp act if it is isomorphic to a subgrou p of some compact Haus d orff group. As usual, w ( X ) denotes the weight of a top ologica l space X . 1. Introduction The classical r esu lt of Hewitt-Marczewski-P ond iczery sta tes: If τ is an infi nite cardin al, I is a set suc h that | I | ≤ 2 τ , and for ev ery i ∈ I a space X i has a d ense sub set of size ≤ τ , then the 1991 Math ematics Subje ct Classific ation. Primary: 22A05; S econdary: 20K99, 22C 05, 54A25, 54 B10, 54D65. Key wor ds and phr ases. ab elian group , monomorphism, homomorphism, p otential ly dense set, dense sub set, pre- compact group. The first named author was partially supp orted by SRA , grants P1-0292-010 1 and J1-9643-010 1. The second named author w as partial ly supp orted by the Gra nt-in-Aid for Scien tific R esearc h (C) No. 19540092 by the Japan So ciety for t he Promotion of Science (JSPS) . 1 2 D. DIKRANJAN AND D. SHAKHM A TOV pro du ct X = Q { X i : i ∈ I } also has a d ense s u bset of size ≤ τ (see, for example, [8, Theorem 2.3.15 ]). In this pap er w e inv estigate the follo wing “algebraic v ersion” of this theorem. Let κ b e an infinite cardin al and T = R / Z the circle group. Giv en a fixed subset S of an ab elian group G , we attempt to find a group homomorph ism π : G → T κ suc h that π ( S ) b ecomes dense in T κ . Of particular int erest is the sp ecial case when π can b e c hosen to b e a monomorphism, that is, when the group G and the su bgroup π ( G ) of T κ b ecome isomorphic. Our c hoice of the target group is justified by the fact that ev ery ab elian group G is isomorphic to a subgroup of T κ for a suitable ca rdinal κ . T o ensure a closer resem b lance of the Hewitt-Marcz ewski-P ondiczery theorem, w e pa y sp ecial at ten tion to the case κ = 2 τ for some infinite cardin al τ by ad d ressing the follo wing question: Giv en a s u bset S of an ab elian group G such that | S | ≥ τ and | G | ≤ 2 2 τ , do es there exist a monomorphism π : G → T 2 τ suc h that π ( S ) b ecomes dense in T 2 τ ? W e completely resolv e this problem in the case when S is un countable. Moreo v er, w e pro vide a complete answer to this problem even for a countable set S suc h that no m ultiple n S of S (for n ∈ N ) is con tained in a finitely generated subgroup of G . T he remaining case is b eing resolv ed b y the authors in [6]. The origin of this setting can b e traced bac k to the 1916 pap er of W eyl [11]. W e r ecall the classical W eyl’s uniform d istribution theorem: Give n a faithfully indexed set S = { a n : n ∈ N } ⊆ Z , the set of all α ∈ T suc h that the set S α = { a n α : n ∈ N } ⊆ T is uniformly distributed h as full measure 1. S ince uniform distribution implies densit y in T , it follo w s that S α is dense in T for almost all α ∈ T . Ev ery α ∈ T determines uniqu ely a homomorph ism h α : Z → T such that h α (1) = α . F urthermore, α ∈ T generate s a dense subgroup h α i of T iff α is non-torsion iff the homomorphism h α is a monomorph ism. Hence, one can state (a consequence of ) W eyl’s theorem b y simply saying that for eve ry in finite subset S of Z , there exists a monomorp hism π : Z → T such that π ( S ) is dense in T . Tk ac henko and Y asc henko [10] consider h omomorphisms π : G → T ω suc h that π ( S ) is dense in T ω , for a certain class of groups G . They u se such homomorphisms as a technical to ol in addressing the problem suggested fi r st in 1946 b y Mark o v [9]. According to Marko v [9], a su bset S of a group G is called p otential ly dense (in G ) provi ded that G admits some Hausdorff group top ology T such that S is dense in ( G, T ). T he last section of [9] is exclusively d edicated to the f ollo wing pr oblem: Whic h subsets of a group G are p oten tially dense in G ? Mark o v prov ed that ev ery infin ite subset of Z is p oten tially dens e in Z [9]. Th is w as strengthened in [7, L emm a 5.2] by sh o wing that eve ry infi nite subset of Z is dens e in s ome precompact metric group topology on Z . (App aren tly , the authors of [9] and [7] w ere u na w are th at b oth these results easily follo w from W eyl’s u n iform distribu tion theorem.) F urther progress w as obtained b y Tk ac henk o and Y asc h enk o [10] who pro ved the follo wing theorem: If an ab elian g roup G of size at most c is either almost torsion-free or has exp onent p for s ome prim e p , then every infinite subset of G is p oten tially dense in G . (Acco rding to [10], an ab elian group G is almost torsion-fr e e if r p ( G ) is finite for ev ery prime p .) In [6], th e authors resolve d Marko v’s problem for coun table sets: A counta ble subs et S of an ab elian group G is p oten tially dense in G if and on ly if | G | ≤ 2 c and S is Zariski d ense in G . Recall that a subset S of an ab elian group G is said to b e Zariski dense in G p r o vided that, if k ∈ N , g 1 , g 2 , . . . , g k ∈ G , n 1 , n 2 , . . . , n k ∈ N and eac h s ∈ S satisfies the equation n i s = g i for some i = 1 , 2 , . . . , k (dep ending on s ), then ev ery g ∈ G also satisfies some equati on n j g = g j , for a suitable j = 1 , 2 , . . . , k ([5]; see also [4, Sectio n 5]). In this manuscript we inv estigate the r emaining case of un coun table sets. In p articular, w e obtain a new sufficien t cond ition that guaran tees that a subs et S of an ab elian group G is p otent ially d ense in G . Moreo ver, w hen this condition is satisfied, we prov e that the top ology T on G su ch that S is T -dense in G can b e chosen to b e precompact. When S is un coun table and G b elongs to a wide class of ab elian groups (for example, almost torsion-free or divisible groups), our sufficient condition tur ns out to b e also necessary for p oten tial density of S in G . A t last but n ot least, our sufficien t condition is rather p o w erful in the countable case as well , b ecuase the only case that is not co v ered by it is when nS , for a suitable n ∈ N , is con tained in a fi nitely generated sub group MARKO V’S POTENTIAL D ENSITY 3 of G . Therefore, it is only this sp ecial case that still requir es the substan tially more soph isticated tec h niques from [6] to p ro v e p otentia l densit y (in some precompact group topology). 2. Sending a given subse t of an abelian group de nsel y in T κ Definition 2.1. Let τ b e an infinite cardinal. W e sa y that a subs et S of an ab elian grou p G is τ -wide if nS \ h S ′ i 6 = ∅ for ev ery n ∈ N and eac h S ′ ⊆ S with | S ′ | < τ . Our next prop osition collects four simp le facts th at clarify the ab o v e d efinition and facilitate future references. Prop osition 2.2. L et S b e a subset of an ab elian gr oup G . (i) If S is unc ountable, then S is τ - wide if and only if min {| nS | : n ∈ N } ≥ τ . (ii) If S c ontains an infinite indep endent sub se t S ′ , then S i s | S ′ | -wide. (iii) If G is torsion, then S i s ω -wide if and only if nS i s infinite for every n ∈ N (that is, S is 0-almost torsion in the sense of [3] ). Ther efor e, G c ontains an ω - wide set S if and only if G is unb ounde d (and in such a c ase the ω - wide sets c oincide with the Zariski dense sets). (iv) If S is not ω -wide, then n S is c ontaine d in a finitely gener ate d sub gr oup of G , for a suitable n ∈ N . Theorem 2.3. L et τ b e an infinite c ar dinal and S a τ - wide subset of an ab elian gr oup G . Then ther e exist a sub gr oup H of G and a monomorphism ϕ : H → T 2 τ such that | H | = τ and ϕ ( H ∩ S ) is dense in T 2 τ . The pro of of this theorem is p ostp oned until Section 5. Corollary 2.4. L et τ b e an infinite c ar dinal and S a τ -wide subset of an ab elian gr oup G . Then: (i) ther e exists a gr oup homomo rphism π : G → T 2 τ such that π ( S ) is dense in T 2 τ ; (ii) if one additional ly assumes that | G | ≤ 2 2 τ , then π fr om the item (i) c an b e chosen to b e a monomor phism. Pr o of. Let H and ϕ b e as in the conclusion of Th eorem 2.3. The pro of n o w branches into t w o cases, d ep ending on w h ic h item of our corollary h olds. (i) Since T 2 τ is a divisible group, there exists a group homomorphism π : G → T 2 τ extending ϕ . (ii) Since | H | = τ < 2 2 τ and r p ( G ) ≤ | G | ≤ 2 2 τ =   T 2 τ   = r p  T 2 τ  for every p ∈ P ∪ { 0 } , w e can extend ϕ to a monomorph ism π : G → T 2 τ , see [3 , Lemma 3.17]. Returning bac k to the common part of the pro of, note th at ϕ ( H ∩ S ) ⊆ π ( S ) and ϕ ( H ∩ S ) is dense in T 2 τ , so π ( S ) m u s t b e d ense in T 2 τ as w ell.  Theorem 2.5. L et κ b e a c ar dinal such that κ > c , and let τ = log κ . F or a subset S of an ab elian gr oup G , the fol lowing c onditions ar e e quivalent: (i) ther e exists a gr oup homomo rphism  : G → T κ such that  ( S ) is dense in T κ ; (ii) τ ≤ min {| nS | : n ∈ N } ; (iii) S is τ -wide. Pr o of. (i) → (ii) Let n ∈ N . Since π ( S ) is dens e in T κ , w ( n T κ ) ≤ 2 | nS | b y Lemma 3.1(ii) b elo w. Since T κ is divisible, n T κ = T κ , and so κ = w ( T κ ) ≤ 2 | nS | , wh ic h yields τ = log κ ≤ | nS | . This pro v es (ii). (ii) → (iii) Since κ > c , τ is uncountable. Applying (ii) w ith n = 1, we conclude that S is uncounta ble. Th en S is τ -wide b y (ii) and Prop osition 2.2(i). (iii) → (i) Let π b e as in the conclusion of Corollary 2.4(i). F rom τ = log κ it follo ws that κ ≤ 2 τ . Let ψ : T 2 τ → T κ b e the pro jection on the first κ co ordinates. Since ψ is a homomorphism , so is  = ψ ◦ π : G → T κ . Since ψ is con tin uous and π ( S ) is d en se in T 2 τ , the set ψ ( π ( S )) =  ( S ) is dense in ψ ( T 2 τ ) = T κ .  4 D. DIKRANJAN AND D. SHAKHM A TOV Theorem 2.6. L et κ b e a c ar dinal such that κ > c , and let τ = log κ . F or a subset S of an ab elian gr oup G , the fol lowing c onditions ar e e quivalent: (i) ther e exists a monomorphism  : G → T κ such that  ( S ) is dense in T κ ; (ii) | G | ≤ 2 κ and τ ≤ min {| nS | : n ∈ N } ; (iii) | G | ≤ 2 κ and S is τ - wide. Pr o of. (i) → (ii) Clearly , | G | = |  ( G ) | ≤ | T κ | = 2 κ . The other inequalit y in item (ii) follo w s f rom the imp licatio n (i) → (ii ) of Theorem 2.5. (ii) → (iii) f ollo ws from the implication (ii) → (ii i) of T h eorem 2.5. (iii) → (i) Let H and ϕ b e as in the conclusion of Th eorem 2.3. F rom τ = log κ it follo ws that τ ≤ κ ≤ 2 τ . F or ev ery h ∈ H \ { 0 } , there exists ξ h ∈ 2 τ suc h that ϕ ( h )( ξ h ) 6 = 0. Define Ξ = { ξ h ∈ 2 τ : h ∈ H \ { 0 }} ⊆ 2 τ . Let µ : T 2 τ → T Ξ b e th e pro jection. Then µ ↾ ϕ ( H ) : ϕ ( H ) → T Ξ is a monomorphism. Since | Ξ | ≤ | H \ { 0 }| ≤ | H | = τ ≤ κ , it follo ws that there exists a homomorph ism ψ : T 2 τ → T κ suc h that χ = ψ ◦ ϕ : H → T κ is a monomorphism. S ince ψ is con tinuous and ϕ ( H ∩ S ) is dense in T 2 τ , the set ψ ( ϕ ( H ∩ S )) = χ ( H ∩ S ) is dense in ψ ( T 2 τ ) = T κ . Since | H | = τ ≤ κ < 2 κ and r p ( G ) ≤ | G | ≤ 2 κ = | T κ | = r p ( T κ ) f or ev ery p ∈ P ∪ { 0 } , w e can extend χ to a monomorph ism  : G → T κ ; s ee [3 , L emma 3.17]. Since χ ( H ∩ S ) ⊆  ( S ) and χ ( H ∩ S ) is d ense in T κ , the set  ( S ) must b e d ense in T κ as well.  The coun ter-part of Theorems 2.5 and 2.6 for κ ≤ c is pro v ed in [6]. 3. Markov’s p otential de nsity W e start this section with a simple necessary condition for p oten tial d en sit y . Lemma 3.1. L et S b e a dense subset of a H ausdorff gr oup G . Then: (i) for every n ∈ N , the set nS is dense in the sub gr oup nG of G ; (ii) w ( nG ) ≤ 2 | nS | and | nG | ≤ 2 2 | nS | for e ach n ∈ N . Pr o of. (i) T he map g 7→ ng that sends G to th e sub group nG of G is con tin uous. Sin ce S is d en se in nG , nS must b e d ense in n G . (ii) Let n ∈ N . F rom (i) we conclude that nS is a dens e sub s et of th e Tyc h onoff sp ace nG . Therefore, w ( nG ) ≤ 2 | nS | b y [8 , Theorem 1.5.7] and | n G | ≤ 2 2 | nS | b y [8, T heorem 1.5.3].  Corollary 3.2. If S is a p otential ly dense subset of a gr oup G , then (3.1) log log | nG | ≤ | nS | for al l n ∈ N . Our next theorem pro vides a general sufficient condition for potentia l density th at also allo ws it to b e realized by some precompact group top ology . Theorem 3.3. L et τ b e an infinite c ar dinal and S a τ -wide subset of an ab elian gr oup G such that | G | ≤ 2 2 τ . Then ther e exist a pr e c omp act Hausdorff gr oup top olo gy T on G such that S is dense in ( G, T ) . Pr o of. Let π : G → T 2 τ b e a monomorph ism from Corollary 2.4(ii), and let T ′ b e the top ology π ( G ) inherits from T 2 τ . Then ( G, T ′ ) is a precompact group . Since π ( S ) is dense in T 2 τ , w e conclude that π ( S ) is T ′ -dense in π ( G ). Since π is an isomorphism b et w een G and π ( G ), T = { π − 1 ( U ) : U ∈ T ′ } is the required top ology .  The c haracterizatio n of the count able p oten tially dense subsets of an ab elian group G with | G | ≤ 2 c can b e foun d in [6]. Our next corollary sh o ws th at Theorem 3.3 is suffi ciently us efu l in obtaining some particular cases of that c haracterizat ion. MARKO V’S POTENTIAL D ENSITY 5 Corollary 3.4. L et G b e an unb ounde d torsion ab elian gr oup with | G | ≤ 2 c . F or a c ountable subset S of G , the fol lowing c onditions ar e e quivalent: (i) S is p otential ly dense in G ; (ii) ther e exist a pr e c omp act H ausdorff gr oup top olo gy T on G such that S is dense in ( G, T ) ; (iii) S is ω -wide. Pr o of. The implication (iii) → (ii) follo w s from Theorem 3.3, while the imp lication (ii) → (i) is trivial. According to Prop osition 2.2(iii), to pro v e the imp licatio n (i) → (iii), it su ffices to c h ec k that nS is infinite for ev ery n ∈ N . Assume that n S is finite for some n ∈ N . By Lemma 3.1(i), th is y ields that nG is finite as w ell. Consequen tly , G is b oun ded, a con tradiction.  In connection with the last corollary , it is worth mentio ning that the u n b ound edness of a torsion ab elian group G is a necessary condition f or the existence of an ω -wide sub set of G ; s ee Prop osi- tion 2.2(iii). Corollary 3.5. L et G b e an ab elian gr oup suc h that | nG | = | G | for every n ∈ N . F or an unc ountable subset S of G , the fol lowing c onditions ar e e quiv alent: (i) S is p otential ly dense in G ; (ii) S is T - dense in some pr e c omp act gr oup top olo gy T on G ; (iii) log log | nG | ≤ | nS | for every n ∈ N ; (iv) log log | G | ≤ min {| nS | : n ∈ N } . Pr o of. The implication (ii) → (i) is obvio us, the implication (i) → (iii) is pr o ved in Corollary 3.2, and the implication (iii) → (iv) h olds due to our assum p tion on G . It r emains only to c hec k the implication (iv) → (ii). F rom (iv) and Prop osition 2.2(i) w e conclude that S is τ -wide for τ = log log | G | . Since | G | ≤ 2 2 τ , Th eorem 3.3 applies.  Corollary 3.6. L et G b e an ab elian gr oup satisfying one of the fol lowing c onditions: (a) G is divisible; (b) | t ( G ) | < | G | ; (c) | t ( G ) | ≤ ω ; (d) G is almost torsion-fr e e. F or every unc ountable sub se t S of G , c onditions (i)–(iv) of Cor ol lary 3.5 ar e e qui valent. Pr o of. Let S b e an uncoun table su bset of G . In particular, | G | ≥ | S | > ω . It suffices to show that G satisfies the assumption of Corollary 3 .5. Fix n ∈ N . If (a) holds, then | nG | = | G | trivia lly h olds, as the homomorph ism η n : G → G defined by η n ( g ) = ng for g ∈ G , is su rjectiv e. I f (b) holds, then from k er η n ⊆ t ( G ) an d | t ( G ) | < | G | it f ollo ws th at | nG | = | G | . I f (c) holds, then t ( G ) ≤ ω < | G | , and so (c) is a particular case of (b). F inally , note that ev ery almost torsion-free group satisfies (c).  Remark 3.7. F or an un coun table subset S of an ab elian group G satisfying items (c) or (d) of Corollary 3.6, the follo wing simplified condition can b e added to the list of equiv alent items (i)–(iv) of Corollary 3.5: (v) log log | G | ≤ | S | . Indeed, s ince t ( G ) is at m ost coun table and S is uncountable, we m ust hav e | nS | = | S | for every n ∈ N . This establishes the implication (v) → (iv) . The rev erse implicat ion (iv) → (v) is tr ivial. Let S b e an uncoun table s u bset of an ab elian group G . Corollaries 3.5 and 3.6 sho w that (3.1) is not only a necessary , bu t also a sufficien t condition for p oten tial dens it y of S in G when G b elongs to a wide class of group s including divisible a nd (a lmost) torsion-free group s . According to Remark 3.7, the w eak er (and muc h simpler) condition log log | G | ≤ | S | , obtained by taking n = 1 in (3.1), is also sufficient for p oten tial densit y of S in G when the torsion part t ( G ) of G is at most 6 D. DIKRANJAN AND D. SHAKHM A TOV coun table. Our next example demonstrates th at this simpler condition (v) is no longer sufficient for p oten tial densit y of S in G when one we ak ens the assum p tion | t ( G ) | ≤ ω to | t ( G ) | < | G | . (Note that | G | ≥ | S | > ω .) Example 3.8. Let Z (2) = Z / 2 Z b e the ab elian group with tw o elements. Define G = Z (2) ω × Z 2 c and S = Z (2) ω × C , where C is an y infinite cyclic subgroup of Z 2 c . Then log log | G | = c = | S | and | t ( G ) | = | Z (2) ω × { 0 }| = c < | G | . Since log log | 2 G | > ω = | 2 S | , the s et S cannot b e p oten tially dense in G b y Corollary 3.2. Observe th at th at a group G w ith the prop erties from Example 3.8 m ust necessarily hav e an uncounta ble to rsion p art t ( G ); s ee Remark 3.7. Example 3.8 pro vides a negativ e answ er to [4, Question 45]. 4. Technical lemmas W e call a connected open subset V of T an op en ar c , and w e use l ( V ) to den ote the length of V . Lemma 4.1. Supp ose that V is an op en ar c in T , z , z ′ ∈ T , m, n ∈ N , 1 ≤ n < m and 2 /m < l ( V ) . Then ther e exists y ∈ V such tha t my = z and ny 6 = z ′ . Pr o of. Since 2 /m < l ( V ), the arc V con tains t w o solutio ns y 0 and y 1 of the equatio n my = z with (4.1) l ( C ( y 0 , y 1 )) = 1 /m, where C ( y 0 , y 1 ) is the shortest arc in T connecting y 0 and y 1 . Supp ose that ny 0 = ny 1 = z ′ . Then n ( y 0 − y 1 ) = 0, and so l ( C ( y 0 , y 1 )) ≥ 1 /n . T oget her with (4.1), th is give s m ≤ n , a con tradiction. Therefore, ny j 6 = z ′ for s ome j = 0 , 1, and so w e can tak e that y j as our y .  If G and H are groups, then G ∼ = H means that G and H are isomorphic. Lemma 4.2. L et τ and κ b e infinite c ar dinals such that τ ≤ κ . L et K b e a sub gr oup of T κ such that | K | ≤ τ . F or ev ery γ < κ , let V γ b e an op en ar c in T such that the set { l ( V γ ) : γ < κ } has a p ositive lower b ound. Then ther e exists f ∈ Q { V γ : γ < κ } having the fol lowing pr op erties: (i) h f i ∼ = Z ; (ii) h f i ∩ K = { 0 } . Pr o of. Fix k ∈ N suc h that 2 /k < l ( V γ ) for ev ery γ < κ . Since | K | ≤ τ , w e can c h o ose an en umeration K × N = { ( h α , n α ) : α < τ } of the s et K × N . By trans finite recursion on α < τ w e will select γ α < κ and y γ α ∈ T satisfying the follo wing prop erties: (i α ) γ α 6∈ { γ β : β < α } , (ii α ) y γ α ∈ V γ α , (iii α ) n α y γ α 6 = h α ( γ α ). Basis of r ecursion . Select γ 0 < κ arb itrarily . Apply Lemma 4.1 to V = V γ 0 , z = 0, z ′ = h 0 ( γ 0 ), n = n 0 and m = n 0 ( k + 1) to c ho ose y γ 0 ∈ T satisfying (ii 0 ) and (iii 0 ). Condition (i 0 ) is v acuous. Recursiv e step . Let α < τ , and su pp ose that γ β < κ and y γ β ∈ T satisfying (i β )–(iii β ) hav e already b een selected for all β < α . W e will n o w c ho ose γ α < κ and y γ α ∈ T satisfying (i α )–(iii α ). Since | α | < τ ≤ κ , w e can c ho ose γ α < κ satisfying (i α ). No w w e apply Lemma 4.1 to V = V γ α , z = 0, z ′ = f α ( γ α ), n = n α and m = n α ( k + 1) to c ho ose y γ α ∈ T satisfying (ii α ), and (iii α ). The recurs ion b eing complete, c ho ose y γ ∈ V γ arbitrarily f or every γ ∈ κ \ { γ α : α < τ } . Define f ∈ T κ b y f ( γ ) = y γ for eac h γ < κ . Then f ∈ Q { V γ : γ < κ } . W e claim that (4.2) nf 6∈ K for ev ery n ∈ N . Indeed, let n ∈ N and h ∈ K b e arbitrary . Then ( h, n ) ∈ K × N , and s o ( h, n ) = ( h α , n α ) for some α < κ . No w nf ( γ α ) = n α y γ α 6 = h α ( γ α ) = h ( γ α ) by (iii α ). Th us, n f 6 = h . Since h ∈ K was arbitrary , this pro v es (4.2). F rom (4.2) we immediately get b oth (i) and (ii).  MARKO V’S POTENTIAL D ENSITY 7 Lemma 4.3. L et τ and κ b e infinite c ar dinals such that τ ≤ κ . L et K b e a sub gr oup of T κ such that | K | ≤ τ . Assume that f ′ ∈ K , m ∈ N and m ≥ 2 . F or every γ < κ , let V γ b e an op en ar c in T such that 2 /m < l ( V γ ) . Then ther e exists f ∈ Q { V γ : γ < κ } satisfying the fol lowing pr op erties: (i) mf = f ′ ; (ii) nf 6∈ K for al l n ∈ N with 1 ≤ n < m . Pr o of. Since | K | ≤ τ , we ca n choose an en um eration K × { 1 , 2 , . . . , m − 1 } = { ( h α , n α ) : α < τ } of the set K × { 1 , 2 , . . . , m − 1 } . By transfinite recur s ion on α < τ w e will select γ α < κ and y γ α ∈ T with the follo wing prop erties: (i α ) γ α 6∈ { γ β : β < α } , (ii α ) y γ α ∈ V γ α , (iii α ) m y γ α = f ′ ( γ α ), (iv α ) n α y γ α 6 = h α ( γ α ). Basis of recursion . Select γ 0 < κ arb itrarily , and apply Lemma 4.1 to V = V γ 0 , z = f ′ ( γ 0 ), z ′ = h 0 ( γ 0 ), n = n 0 and m to c ho ose y γ 0 ∈ T satisfying (ii 0 ), (iii 0 ) and (iv 0 ). Condition (i 0 ) is v acuous. Recursiv e step . Let α < τ , and supp ose that γ β < κ and y γ β ∈ T satisfying (i β )–(iv β ) hav e already b een selected for all β < α . W e will now c ho ose γ α < κ and y γ α ∈ T satisfying (i α )–(iv α ). Since | α | < τ ≤ κ , w e can c ho ose γ α < κ satisfying (i α ). No w w e apply Lemma 4.1 to V = V γ α , z = f ′ ( γ α ), z ′ = h α ( γ α ), n = n α and m to c h o ose y γ α ∈ T satisfying (ii α ), (iii α ) and (iv α ). The recursion b eing complete, for ev ery γ ∈ κ \ { γ α : α < τ } , apply Lemma 4.1 to V = V γ , z = f ′ ( γ ), z ′ = 0, n = 1 and m to choose y γ ∈ V γ suc h that my γ = f ′ ( γ ). Define f ∈ T κ b y f ( γ ) = y γ for ev ery γ < κ . Th en f ∈ Q { V γ : γ < κ } and (i) is satisfied. T o prov e (ii), c ho ose n ∈ N suc h that 1 ≤ n < m . Let h ∈ K b e arbitrary . T hen ( h, n ) ∈ K × { 1 , 2 , . . . , m − 1 } , and so ( h, n ) = ( h α , n α ) f or s ome α < κ . No w nf ( γ α ) = n α y γ α 6 = h α ( γ α ) = h ( γ α ) by (iv α ). Therefore, nf 6 = h . Sin ce h ∈ K was arbitrary , this pro v es n f 6∈ K .  Lemma 4.4. L et G and G ∗ b e ab elian gr oups, and let K and K ∗ b e su b gr oups of G and G ∗ , r esp e ctive ly. Supp ose also tha t x ∈ G , x ∗ ∈ G ∗ , m ∈ N , m ≥ 2 , and ψ : K → K ∗ is an isomorphism satisfying the fol lowing pr op erties: (a) mx ∈ K and mx ∗ ∈ K ∗ ; (b) nx 6∈ K and nx ∗ 6∈ K ∗ whenever n ∈ N and 1 ≤ n < m ; (c) ψ ( mx ) = mx ∗ . Then ther e exists a unique isomorp hism ϕ : K + h x i → K ∗ + h x ∗ i extending ψ such that ϕ ( x ) = x ∗ . Pr o of. Define ϕ b y (4.3) ϕ ( h + k x ) = ψ ( h ) + k x ∗ for h ∈ K and k ∈ Z . T o c hec k that this d efinition is correct, supp ose that (4.4) h + k x = h ′ + k ′ x, with h, h ′ ∈ K and k, k ′ ∈ Z . Hence, ( k ′ − k ) x = h − h ′ ∈ K , so by (a) and (b), one h as k ′ − k = l m for some l ∈ Z , wh ich yields h − h ′ = lmx . T his giv es ψ ( h ) − ψ ( h ′ ) = ψ ( h − h ′ ) = ψ ( l mx ) = l ψ ( mx ) = l mx ∗ = ( k ′ − k ) x ∗ = k ′ x ∗ − k x ∗ b y (c), and so ψ ( h ) + k x ∗ = ψ ( h ′ ) + k ′ x ∗ . Comparing this with (4.4), we conclude that (4.3) correctly defines a homomorphism ϕ . F r om (4.3) we get ϕ ( x ) = x ∗ and ϕ ↾ K = ψ . Moreo ver, ϕ is surjectiv e and unique with these prop erties. T o p ro v e that ϕ is a m on omorp hism, assume that ϕ ( h + k x ) = 0 f or some h ∈ K and k ∈ Z . Then ψ ( h ) + k x ∗ = 0, so k x ∗ ∈ K ∗ . C onsequen tly , m divides k by (a) and (b). Then k x ∈ K , 8 D. DIKRANJAN AND D. SHAKHM A TOV and so h + k x ∈ K as well. Therefore, ϕ ( h + k x ) = ψ ( h + k x ) = 0 yields h + k x = 0, as ψ is a monomorphism.  5. Proof of Theorem 2.3 Fix a counta ble b ase V for the top ology of T consisting of op en arcs of T su ch th at T ∈ V . Consider th e T ychonoff pro d u ct top ology on 2 τ , and let B b e the canonical b ase for 2 τ consisting of n on -emp t y clop en sub sets of 2 τ suc h that | B | = τ . Let U =  U ∈ [ B ] <ω : U is a co v er of 2 τ b y pairwise disjoin t sets  . F or ξ ∈ 2 τ and U ∈ U , let U ξ , U ∈ U denote the uniqu e U ∈ U suc h that ξ ∈ U . Define E = { ( U , v ) : U ∈ U and v : U → V is a function } . F or ( U , v ) ∈ E , let F ( U , v ) = n f ∈ T 2 τ : f ( ξ ) ∈ v ( U ξ , U ) for all ξ ∈ 2 τ o = Y  v ( U ξ , U ) : ξ ∈ 2 τ  . Clearly , | E | = τ , so w e can fix an enumeration E = { ( U α , v α ) : α < τ } of E suc h that U 0 = { 2 τ } and v 0 (2 τ ) = T . F or eac h α < τ , c ho ose n α ∈ N such that (5.1) 2 / n α < min { l ( v ( U )) : U ∈ U } . By transfinite r ecursion on α < τ we will c h o ose an element x α ∈ S and define a map ϕ α : H α = h{ x β : β ≤ α }i → T 2 τ satisfying the follo wing conditions: (i α ) ϕ α ( x α ) ∈ F ( U α , v α ), (ii α ) ϕ α is a m onomorphism, (iii α ) ϕ α ↾ H β = ϕ β for all β < α . Basis of recursion . Pic k x 0 ∈ S arbitrarily , and let ϕ 0 : h x 0 i → T 2 τ = F ( U 0 , v 0 ) b e an arbitrary monomorphism. Th en conditions (i 0 ) and (ii 0 ) are satisfied, while the condition (iii 0 ) is v acuous. Recursiv e step . Let α < τ , and supp ose that x β ∈ S and a map ϕ β : H β → T 2 τ satisfying (i β ), (ii β ) and (iii β ) hav e already b een constructed for ev ery β < α . W e are going to d efine x α ∈ S and a map ϕ α : H α → T 2 τ satisfying (i α ), (ii α ) and (iii α ). Define H ′ α = h{ x β : β < α }i = [ β <α H β . Since (ii β ) and (iii β ) hold for ev ery β < α , ϕ ′ α = [ β <α ϕ β : H ′ α → T 2 τ is a monomorphism. Since { x β : β < α } ⊆ S , | α | < τ and S is τ -wide, one has n α ! S \ h{ x β : β < α }i = n α ! S \ H ′ α 6 = ∅ , and so there exists x α ∈ S suc h that n α ! x α 6∈ H ′ α . In par- ticular, (5.2) nx α 6∈ H ′ α for all n ≤ n α . Let κ = 2 τ and K = ϕ ′ α ( H ′ α ). F or ξ ∈ 2 τ , define V ξ = v α ( U ξ , U α ). Then (5.1 ) yields (5.3) 2 /n α < l ( v α ( U ξ , U α )) = l ( V ξ ) fo r every ξ ∈ 2 τ . W e need to consider tw o cases. Case 1 . { n ∈ N \ { 0 } : nx α ∈ H ′ α } = ∅ . In this case, h x α i ∼ = Z and the sum h x α i + H ′ α = h x α i ⊕ H ′ α is direct. S ince { l ( V γ ) : γ < κ } has a p ositiv e lo wer b ound by (5.3), we can apply Lemma 4.2 to c ho ose (5.4) f ∈ Y { V ξ : ξ ∈ 2 τ } = F ( U α , v α ) MARKO V’S POTENTIAL D ENSITY 9 with h f i ∼ = Z and h f i ∩ K = { 0 } . Th en the su m K + h f i = K ⊕ h f i is direct as wel l. Since h x α i ∼ = h f i ∼ = Z , there exists a u n ique monomorphism ϕ α : H α = H ′ α ⊕ h x α i → K ⊕ h f i ⊆ T 2 τ extending ϕ ′ α suc h that ϕ α ( x α ) = f . Case 2 . { n ∈ N : nx α ∈ H ′ α } 6 = ∅ . Let m = min { n ∈ N : nx α ∈ H ′ α } and f ′ = ϕ ′ α ( mx α ) ∈ K . Then m > n α b y (5.2), so from (5.3) w e obtain 2 /m < 2 /n α < l ( V ξ ) for ev ery ξ ∈ 2 τ . Since n α ≥ 1, w e ha v e m ≥ 2. App lying Lemma 4 .3, we get f satisfying (5.4) such that mf = f ′ and nf 6∈ K for all n ∈ N with 1 ≤ n < m . Ob serv e that T 2 τ (tak en as G ∗ ), H ′ α (tak en as K ), K (tak en as K ∗ ), x α (tak en as x ), f (tak en as x ∗ ), ϕ ′ α (tak en as ψ ) and m satisfy th e assum ptions of L emm a 4.4. Denoting ϕ from the conclusion of this lemma by ϕ α , w e obtain a monomorphism ϕ α : H α → T 2 τ extending ϕ ′ α suc h that ϕ α ( x α ) = f . The monomorphism ϕ α ob viously satisfies (i α ), (ii α ) and (iii α ) in b oth cases. The recursive construction b eing complete, let H = [ α<τ H α and ϕ = [ α<τ ϕ α . Since (ii α ) and (iii α ) are satisfied for ev ery α < τ , ϕ : H → T 2 τ is a m onomorphism. It remains only to c hec k that ϕ ( H ∩ S ) is dense in T 2 τ . Let W b e a non-emp ty op en subset of T 2 τ . Cho ose pairwise distinct ξ 1 , . . . , ξ i ∈ 2 τ and V 1 , . . . , V i ∈ V suc h that n f ∈ T 2 τ : f ( ξ j ) ∈ V j for all j ≤ i o ⊆ W . Select U = { U 1 , . . . , U i } ∈ U w hic h separates ξ j ’s; that is, j, k ≤ i and ξ j ∈ U k implies j = k . Define v : U → V b y v ( U j ) = V j for j ≤ i . Th en ( U , v ) ∈ E , and so ( U , v ) = ( U α , v α ) for some α < τ . Note th at (5.5) F ( U α , v α ) = F ( U , v ) ⊆ n f ∈ T 2 τ : f ( ξ j ) ∈ V j for all j ≤ i o ⊆ W . Since x α ∈ S ∩ H α ⊆ H ∩ S an d ϕ ( x α ) = ϕ α ( x α ) ∈ F ( U α , v α ) by (i α ), it follo ws from (5.5 ) that ϕ ( x α ) ∈ ϕ ( H ∩ S ) ∩ W 6 = ∅ . 6. Questions All instances of p otenti al densit y in Section 3 w ere witnessed b y a precompact Hausdorff group top ology . T his mak es it natural to ask the follo wing Question 6.1. Let S b e a p oten tially d ense subset of an ab elian group G . Do es there exist a Hausdorff pr ecompact group top ology T on G s u c h that S is T -dense in G ? The answe r to this question is p ositiv e when S is coun table [6]. It w as sho wn in [5] that ev ery p oten tially dense subset S of an ab elian group G must b e Zariski dense in G . (W e refer the reader to the In tro duction for the definition of Zariski densit y .) Corol- lary 3.2 giv es another necessa ry condition for p oten tial density of S in G . On e ma y w ond er if these t w o cond itions, com b ined together, are also s ufficien t: Question 6.2. L et S b e an infinite sub set of an Ab elian group G suc h that: (a) S is Zariski dense in G , and (b) log log | nG | ≤ | nS | (equiv alen tly , | nG | ≤ 2 2 | nS | ) for all n ∈ N . Is S p oten tially dense in G ? Do es there exist a Hausdorff pr ecompact group top ology T on G suc h that S is T -dense in G ? W e note th at this question is an appropr iate m o dification of [4, Qu estion 45] that is necessary in view of Example 3.8. 10 D. DIKRANJAN AND D. SHAKHM A TOV A cknowledgment It is our pleasure to thank the referee f or her/his useful suggestions that inspired the authors to re-design S ection 3. Referen ces [1] D. N. Dikranjan, I. R. Prodanov, and L. N. Sto yano v , T op olo gi c al Gr oups (Char acters, Dual ities and Mi nimal Gr oup T op olo gies) , Monographs and T extb o oks in Pure and App lied Mathematics 130 , Marcel Dekker, Inc., New Y ork-Basel, 1990. [2] D. Dikranjan and D. Sh ak hmato v, Algebr aic str uctur e of pseudo c omp act gr oups , Mem. Amer. Math. S oc. 133 (1998), 83 pages. [3] D. Dikranjan and D. Shakh matov, F or cing her e di tarily sep ar able c omp act-like gr oup top olo gies on ab elian gr oups , T op ology A ppl. 151 (2005), 2–54. [4] D. Dikranjan and D . Shakh mato v, Sele cte d topics fr om the structur e the ory of top olo gic al gr oups , pp. 389-406 in: Op en Problems in T op ology II (E.Perl , ed.), El sevier 2007. [5] D. Dikranjan and D. Sh akhmato v, The Markov- Zariski top olo gy of an ab elian gr oup , submitted. [6] D. Dikranjan and D. Sh akhmato v, A Kr one cker-Weyl the or em for subse ts of A b elian gr oups , w ork in progress. [7] D. Dikranjan and M. Tk aˇ cenko, We akly c omplete fr e e top olo gic al gr oups , T op ology Ap p l. 112 (2001), 259– 287. [8] R. Engel’king, General t opology (Second ed ition), S igma Series in Pure Mathematics, 6 , H eldermann V erlag, Berlin, 1989. [9] A. A. Marko v, On unc onditional ly close d sets , Mat. Sb ornik 18 (1946), 3–28 (in Russian); English translation in: A. A. Mark ov, Three pap ers on top ologica l groups: I. On the existence of p erio dic connected t op ological groups. I I. On free topological groups. I I I. On un conditionally clo sed sets, Amer. Math. Soc. T ranslation 1950, (1950 ). no. 30, 120 pp .; y et another English translation in: T op ology and T op ological A lgebra, T ranslations Series 1, vol . 8, pp. 273–304, A mer. Math. So c., 1962. [10] M. Tk ac henko and I. Y asc henko, Indep endent gr oup top olo gies on ab elian gr oups , T op ology Appl. 122 (2002), 425–451 . [11] H. W eyl, ¨ Ub er die Gleichver teilung von Zahlen mo d. Eins (in German), Math. A n n. 77 (1916), no. 3, 313–352. Universit ` a di Udine, Dip ar timen to di Ma tema tica e Informa tica, via delle Scienze, 206 - 33100 Udine, It al y E-mail addr ess : dikran.dik ranjan@dimi. uniud.it Gradua te S ch ool of S cience and Engi neering, Division of Ma thema tics, Physics a nd Ear th Sciences, Ehime University, Ma tsuy ama 790-8 577, Jap an E-mail addr ess : dmitri@dpc .ehime-u.ac. jp