Constructing universally small subsets of a given packing index in Polish groups

CONSTRU CTING UNIVERSALL Y SMALL SUBSETS OF A GIVEN P A CKING INDEX IN POLISH GR OUPS T ARAS BANAKH AND NADY A L Y ASKOVSKA Abstra ct. A subset o f a P olish space X is called universal ly smal l if i t b elongs to each ccc σ - ideal with Borel base on X . Under CH in ea c h uncountable Ab elian Polish group G we construct a univ ersally small subset A 0 ⊂ G such that | A 0 ∩ g A 0 | = c for e ac h g ∈ G . F or eac h cardinal num ber κ ∈ [5 , c + ] the set A 0 conta ins a universally sma ll subset A of G with sharp packing index pack ♯ ( A κ ) = sup {|D| + : D ⊂ { g A } g ∈ G is disjoin t } equal to κ . 1. Introduction This pap er is motiv ated b y a p r oblem of Dikranjan and Protaso v [4] who ask ed if the group of in tegers Z conta ins a su bset A ⊂ Z su c h that the family of sh if ts { x + A } x ∈ Z con tains a d isjoin t subfamily of arbitrarily large fi nite cardinalit y but do es not conta in s an infinite disjoint subfamily . This problem can b e reform ulated in the language of pac king in dices pac k( A ) and pac k ♯ ( A ), defin ed for any subset A of an group G b y the form ulas: pac k ( A ) = sup {|D | : D ⊂ { g A } g ∈ G is a disjoint subfamily } and pac k ♯ ( A ) = s u p {|D| + : D ⊂ { g A } g ∈ G is a disjoint subfamily } . So, act u ally Dikranjan and Protaso v ask ed ab out the existence of a subset A ⊂ Z with pac king index pac k ♯ ( A ) = ℵ 0 . Th is problem was answered affirmativ ely in [1 ] and [2]. Moreo v er, in [7 ] the second author pro ve d that for an y cardinal κ with 2 ≤ κ ≤ | G | + and κ / ∈ { 3 , 4 } in any Ab elian group G there is a subset A ⊂ G with pac king in dex pac k ♯ ( A ) = κ . By Theorem 6.3 of [3], su c h a set A can b e fou n d in any sub set L ⊂ A with Pac king in dex Pa c k( L ) = 1 wh ere P ac k ( A ) = sup {|A| : A ⊂ { g A } g ∈ G is | G | -almost d isjoin t } . A family A of s ets is called κ -almost disjoint for a cardinal κ if | A ∩ A ′ | < κ for an y distinct sets A, A ′ ∈ A . S o, b eing 1-almost disjoint is equiv alen t to b eing disjoint. A subset A ⊂ G with s m all pac kin g index can b e thought as large in a geometric sense b ecause in this case th e group G d oes not conta in many disjoin t translation copies of A . It is natural to compare this largeness p rop ert y with other largeness prop erties that ha ve top olog ical or measure- theoretic nature. It turns out that a set of a group A can hav e small pac king index (so can b e large in geometric sense) and simultaneously b e small in other sen s es. In [3] it w as pro v ed that eac h uncounta b le Po lish Ab elian group G contai n s a closed su bset A ⊂ G th at has large pac kin g index P ac k ( A ) = 1 but is nowhere dens e and Haar null in G . According to Th eorem 16.3 [9], und er CH (the Con tinuum Hyp othesis), eac h Po lish group G con tains a sub set A with pac king ind ex pac k ( A ) = 1, whic h is universal ly nul l in the sense that A has m easure zero with r esp ect to any atomless Borel probabilit y measure on G . In this pap er w e mov e further in this d ir ecti on and pro ve that under 1991 Mathematics Subje ct C l assific ation. 03E15; 03E50; 22A05; 54H05; 54H11. Key wor ds and phr ases. Universall y small set, universal ly meager set, unive rsally null set, pac king index, P olish group, coanalytic set. 1 2 T ARAS BANAKH AND NADY A L Y ASKO VSKA CH eac h uncounta ble Ab elian Polish group G con tains a subset A ⊂ G w ith large pac king ind ex P ac k ( A ) = 1, whic h is universal ly smal l in the sense that it b elongs to an y ccc Borel σ -ideal on G . This fact com bined with T heorem 6.3 of [3] allo ws us to construct unive rsally small sub s ets of a giv en pac king index in un coun table Polish Ab elian groups. F ollo wing Zakrzewski [11] w e call a subset A of a Polish space X universal ly smal l if A b elongs to eac h ccc σ -ideal with Borel base on X . By an ide al on a set X w e un derstand a family I of subsets of X suc h that • ∪I = X / ∈ I ; • for any sets A, B ∈ I we get A ∪ B ∈ I ; • for any sets A ∈ I and B ⊂ X w e get A ∩ B ∈ I . An ideal I on a P olish space X is called • a σ -ide al if ∪A ∈ I for any coun table sub f amily A ⊂ I ; • an ide al with Bor el b ase if eac h set A ∈ I lies in a Borel subs et B ∈ I ; • a c c c ide al if X con tains no uncountable disj oin t family of Borel subs ets outside I . Standard examples of ccc Borel σ -ideals are th e ideal M of meager su bsets of a Po lish space X and the id eal N of n ull sub sets w ith resp ect to an atomless Borel σ -additiv e measur e on X . This implies that a universally small su bset A is u niv ersally null and unive r sally meager. F ollo wing [10] we call a subset A of a Po lish space X to b e universal ly me ager if for an y Borel isomorph ism f : A → 2 ω the image f ( A ) is meager in the Canto r cub e 2 ω . Universally small sets w ere in tro duced b y P .Zakrzewski [11] who constru cted an uncounta ble un iv ersally small s ubset in eac h uncoun table P olish space. It should b e mentio n ed that there are mo dels of ZFC [8, § 5] in whic h all universally small sets in P olish spaces ha ve cardinalit y ≤ ℵ 1 < c . In such mo dels an y unive r sally sm all set A in the real line has maximal p ossible pac king ind ex pac k( A ) = Pa ck( A ) = c . This fact sh o ws that the follo win g theorem, whic h is the main r esult of this p ap er , necessarily has consistency nature and cannot b e pro ved in ZF C. Theorem 1. Under CH, e ach unc ountable Ab elian Polish gr oup G c ontains a universal ly smal l sub set A 0 ⊂ G with Packing index Pa ck( A 0 ) = 1 . Com bin ing this th eorem with Theorem 6.4 of [3] w e get Corollary 1. Under CH, for any c ar dinal κ ∈ [2 , c + ] with κ / ∈ { 3 , 4 } any unc ountable P olish gr oup G c ontains a universal ly smal l sub se t A with sharp p acking index pack ♯ ( A ) = κ . 2. Universall y small sets from coanal ytic rank s In this section we describ e a (known) metho d of constructing un iv ersally small sets, based on coanalytic ranks. Let us r ecal l that a sub set A of a Poli sh sp ace X is • analytic if A is the con tinuous image of a Poli sh sp ace ; • c o analytic if X \ A is analytic. By Souslin’s Th eorem [5, 14.11], a sub s et of a Pol ish space is Borel if and only if it is analytic an d coanalytic . It is kn o wn [5, 34.4] that eac h coanalytic su b set K of a P olish space X admits a r ank function rank : K → ω 1 that has the follo win g prop erties: (1) for ev ery countable ordin al α th e set B α = { x ∈ K : rank( x ) ≤ α } is Borel in X ; (2) eac h analytic su bspace A ⊂ K lies in some set B α , α < ω 1 . P ACKING INDEX OF UNIVERSALL Y SM ALL SETS IN POLISH GROUPS 3 The follo w ing fact is kno w n and b elongs to mathematica l folklore (cf. [8, 5.3]). F or the con v en ience of the reader we attac h a sh ort pro of. Lemma 1. L et K b e a c o analytic non-analytic set in a Polish sp ac e X and r an k : K → ω 1 b e a r ank function. F or any tr ansfinite se quenc e of p oints x α ∈ K \ B α , α ∈ ω 1 , the set { x α } α ∈ ω 1 is u niversal ly smal l in X . Pr o of. Giv en an y ccc Borel σ -ideal I on X , use th e classical Szpilra jn-Marczewski Theorem [6, § 11] to conclude that th e coanalytic s et K b elongs to the completion B I ( X ) = { A ⊂ X : ∃ B ∈ B ( X ) A △ B ∈ I } of the σ -algebra of Borel subsets of X b y the ideal I . Consequen tly , ther e is a Borel subset B ⊂ K of X such that K \ B ∈ I . By the prop erty of the rank function, the Borel set B lies in B β for some counta b le ordinal β . Then the set { x α } α<ω 1 b elongs to the σ -ideal I , b eing the union of th e coun table set { x α } α ≤ β and the set { x α } β <α<ω 1 ⊂ K \ B α ⊂ K \ B from the id eal I .  In ord er to p ro v e T heorem 1 w e shall com b ine Lemma 1 with the follo wing tec h nical lemma that will b e pro v ed in Section 4. Lemma 2. F or any unc ountable P olish Ab elian gr oup G ther e ar e a non-empty op en se t U ⊂ G and a c o-analytic subset K of G such that U ⊂ ( K \ A ) − ( K \ A ) for any analytic subsp ac e A ⊂ K of G . 3. Proof of Theorem 1 Assume the Con tinuum Hyp othesis. Giv en an u n coun table P olish Ab elian group G w e n eed to construct a univ ersally sm all su bset A ⊂ G with P ac k ( A ) = 1. W e shall use the add itiv e notatio n for denoting the group op eration on G . So, 0 will denote the neutral element of G . F or t wo sub sets A, B ⊂ G we p ut A + B = { a + b : a ∈ A, b ∈ B } and A − B = { a − b : a ∈ A, b ∈ B } . By Lemma 2, there are a non -emp t y op en set U ⊂ G and a coanalytic subset K suc h that U ⊂ ( K \ B ) − ( K \ B ) for any Borel su b set B ⊂ K of G . This imp lies that the coanalytic set K is n ot Borel in G . Let r ank : K → ω 1 b e a rank f unction for K . T his fu n ctio n ind uces the decomp osition K = S α<ω 1 B α in to Borel sets B α = { x ∈ K : rank( x ) ≤ α } , α < ω 1 , suc h that eac h Borel sub set B ⊂ K of G lies in some set B α , α < ω 1 . The Con tinuum Hyp othesis allo ws u s to c ho ose an en u meration U = { u α } α<ω 1 of the op en s et U suc h that for ev er y u ∈ U th e set Ω u = { α < ω 1 : u α = u } is uncountable. The sep arabilit y of G yields a counta ble sub set C ⊂ G s uc h that G = C + U . By ind uction, for ev ery α < ω 1 find t wo p oint s x α , y α ∈ K \ ( B α ∪ { x β : β < α } ) such that x α − y α = u α . S u c h a c hoice is alwa ys p ossible as U ⊂ ( K \ B ) − ( K \ B ) for an y Borel subs et B ⊂ K of G . Lemma 1 guarante es that the s ets { x α } α<ω 1 and { y α } α<ω 1 are universally small in G and so is the set A = { c + x α , y α : c ∈ C , α < ω 1 } . It remains to p r o v e that Pac k ( A ) = 1. This equalit y will follo w as so on as we c heck that for every p oint z ∈ G the intersectio n A ∩ ( z + A ) has cardinalit y of conti nuum. Since C + U = G , we can fin d elemen ts c ∈ C and u ∈ U su c h that z = c + u . The c hoice of the en um eratio n { u α } α<ω 1 guaran tees that the set Ω u = { α < ω 1 : u α = u } has card inalit y con tinuum. No w observe that f or ev ery α ∈ Ω u w e get z = c + u = c + u α = c + x α − y α and hen ce c + x α = z + y α ∈ A ∩ ( z + A ), wh ic h implies th at the in tersection A ∩ ( z + A ) ⊃ { c + x α } α ∈ Ω u has cardinalit y of con tin uu m. 4. Pr oof of Lemm a 2 Fix an inv arian t metric d ≤ 1 generating the top ology of G . This metric is complete b ecause the group G is P olish. Th e m etric d induces a norm k · k : G → [0 , 1] on G defined by k x k = d ( x, 0). F or 4 T ARAS BANAKH AND NADY A L Y ASKO VSKA an ε > 0 by B ( ε ) = { x ∈ G : k x k < ε } and ¯ B ( ε ) = { x ∈ G : k x k ≤ ε } we shall denote the op en and closed ε -balls cent ered at zero. W e defin e a subset D of G to b e ε -sep ar ate d if d ( x, y ) ≥ ε for an y d istinct p oints x, y ∈ D . By Zorn’s Lemma, eac h ε -separated su bset S of an y su bset A ⊂ G can b e enlarged to a maximal ε -separated subset ˜ S of A . This set ˜ S is ε -net for A in the sense th at for eac h p oin t a ∈ A there is a p oint s ∈ ˜ S with d ( a, s ) < ε . Fix an y non-zero elemen t a − 1 ∈ G and let ε − 1 = 1 12 k a − 1 k . By indu ction w e can d efi ne a sequence ( ε n ) n ∈ ω of p ositiv e r eal num b ers and a sequence ( a n ) n ∈ ω of p oints of the group G s uc h that • 16 ε n ≤ k a n k < ε n − 1 for ev ery n ∈ ω . F or ev ery n ∈ ω , fi x a maximal 2 ε n -separated su b set X n ∋ 0 in the ball B (2 ε n − 1 ). The choic e of the sequence ( ε n ) guarantee s that the series P n ∈ ω ε n is con vergen t and th u s for an y sequence ( x n ) n ∈ ω ∈ Q n ∈ ω X n the series P n ∈ ω x n is con verge nt in G (b ecause k x n k < 2 ε n − 1 for all n ∈ N ). T herefore the follo wing sub sets of the group G are well- defined: Σ 0 =  X n ∈ ω x 2 n : ( x 2 n ) n ∈ ω ∈ Y n ∈ ω X 2 n  , Σ 1 =  X n ∈ ω x 2 n +1 : ( x 2 n +1 ) n ∈ ω ∈ Y n ∈ ω X 2 n +1  . These sets ha v e the follo wing prop erties: Claim 1. (1) Σ 0 ∪ Σ 1 ⊂ B (4 ε − 1 ) . (2) B (2 ε − 1 ) ⊂ Σ 1 + Σ 0 . (3) F or e v ery i ∈ { 0 , 1 } the closur e Σ i − Σ i of the set Σ i − Σ i in G is not a neighb orho o d of zer o. Pr o of. 1. F or every p oin t x ∈ Σ 0 ∪ Σ 1 w e can fi nd a s equ ence ( x n ) n ∈ ω ∈ Q n ∈ ω X n with x = P ∞ n =0 x n and observe that k x k ≤ ∞ X n =0 k x n k ≤ ∞ X n =0 2 ε n − 1 < X n ∈ ω 2 ε − 1 16 n < 4 ε − 1 . 2. Giv en any p oin t x ∈ B (2 ε − 1 ), find a p oin t x 0 ∈ X 0 suc h that k x − x 0 k < 2 ε 0 . Such a p oin t x 0 exists as the set X 0 is a 2 ε 0 -net in B (2 ε − 1 ). Contin uing b y indu ctio n , for ev ery n ∈ ω find a p oin t x n ∈ X n suc h that k x − P n i =0 x i k < 2 ε n . After completing the inductiv e constru ctio n , we obtain a sequence ( x n ) n ∈ ω ∈ Q n ∈ ω X n suc h th at x = X n ∈ ω x n = X n ∈ ω x 2 n + X n ∈ ω x 2 n +1 ∈ Σ 0 + Σ 1 . 3. W e shall give a detail pro of of th e third statemen t f or i = 0 (for i = 1 the pro of is analogous). Since the sequence ( a 2 k + 1 ) k ∈ ω con ve r ges to zero, it suffices to sh o w that d ( a 2 k + 1 , Σ 0 − Σ 0 ) > 0 for all k ∈ ω . Giv en t w o p oints x, y ∈ Σ 0 , we shall pro ve that d ( a 2 k + 1 , x − y ) ≥ ε 2 k + 1 . If x = y , then d ( a 2 k + 1 , x − y ) = d ( a 2 k + 1 , 0) = k a 2 k + 1 k > ε 2 k + 1 b y the c hoice of a 2 k + 1 . So, we assume that x 6 = y . Fin d infinite sequences ( x 2 n ) n ∈ ω , ( y 2 n ) n ∈ ω ∈ Q n ∈ ω X 2 n with x = P n ∈ ω x 2 n and y = P n ∈ ω y 2 n . Let m = min { n ∈ ω : x 2 n 6 = y 2 n } . If m ≥ k + 1, then k x − y k = k X n ≥ m x 2 n − y 2 n k ≤ X n ≥ m k x 2 n k + k y 2 n k ≤ ≤ 2 X n ≥ m 2 ε 2 n − 1 ≤ 8 ε 2 m − 1 ≤ 8 ε 2 k + 1 < k a 2 k + 1 k − ε 2 k + 1 P ACKING INDEX OF UNIVERSALL Y SM ALL SETS IN POLISH GROUPS 5 and hence d ( x − y , a 2 k + 1 ) ≥ ε 2 k + 1 . If m ≤ k , then k x − y k = k ( x 2 m − y 2 m ) + X n>m ( x 2 n − y 2 n ) k ≥ k x 2 m − y 2 m k − X n>m ( k x 2 n k + k y 2 n k ) ≥ ≥ 2 ε 2 m − 2 X n>m 2 ε 2 n − 1 ≥ 2 ε 2 m − 8 ε 2 m +1 ≥ 3 2 ε 2 m ≥ 3 2 ε 2 k > k a 2 k + 1 k + 1 2 ε 2 k according to the choice of the p oin t a 2 k + 1 . Consequently , d ( x − y , a 2 k + 1 ) ≥ 1 2 ε 2 k ≥ ε 2 k + 1 .  A s u bset C of G will b e called a Cantor set in G if C is h omeomorph ic to th e Can tor cub e { 0 , 1 } ω . By the classical Brouw er’s Theorem [5, 7.4], this happ ens if and only if C is compact, zero-dimensional and has no isolated p oints. Claim 2. F or every i ∈ { 0 , 1 } ther e is a Cantor set C i ⊂ B ( ε 0 ) such that the map h i : C i × Σ i → G , h i : ( x, y ) 7→ x + y , is a close d top olo gic al emb e dding. Pr o of. T aking into accoun t that Σ i − Σ i = Σ i − Σ i is not a neigh b orh oo d of zero in G , and rep eating the p r oof of Lemma 2.1 of [3], w e can construct a C an tor set C i ⊂ B ( ε 0 ) suc h that for any d istinct p oin ts x, y ∈ C i the shifts x + Σ i and y + Σ i are d isjoin t. T h is implies that the map h i : C i × Σ i → G , h i : ( x, y ) 7→ x + y , is injectiv e. T aking into accoun t that th e set C i is compact and Σ i is closed in G , one can c hec k that the map h i is closed and h en ce a closed top ologica l em b eddin g.  Observe that for ev er y i ∈ { 0 , 1 } the em b edding h i has image h i ( C i × Σ i ) = C i + Σ i ⊂ B ( ε 0 ) + ¯ B (4 ε − 1 ) ⊂ B (5 ε − 1 ). No w w e mo dify th e closed em b eddings h 0 and h 1 to closed em b edd ings ˜ h 0 : C 0 × Σ 0 → G, ˜ h 0 : ( x, y ) 7→ a − 1 + x + y and ˜ h 1 : C 1 × Σ 1 → G, ˜ h 1 ( x, y ) = − x − y . These embedd ings ha v e images ˜ h 0 ( C 0 × Σ 0 ) ⊂ a − 1 + B (5 ε − 1 ) and ˜ h 1 ( C 1 × Σ 1 ) ⊂ a − 1 − B (5 ε − 1 ) = a − 1 + B (5 ε − 1 ). Since k a − 1 k = 12 ε − 1 , we conclude that the closed sub sets ˜ h i ( C i × Σ i ), i ∈ { 0 , 1 } , of G are disjoint. F or ev ery i ∈ { 0 , 1 } fix a coanalytic n on -analytic su bset K i in the Cantor set C i . It follo ws th at the disjoint un ion K = S 1 i =0 ˜ h i ( K i + Σ i ) is a coanalytic subset of G . The follo wing claim completes th e p ro of of the lemma and sh o w s th at the coanalytic set K and the op en set U = a − 1 + B ( ε − 1 ) ha ve the required prop ert y . Claim 3. U ⊂ ( K \ A ) − ( K \ A ) for any analytic subsp ac e A ⊂ K . Pr o of. Giv en an analytic sub s pace A ⊂ K , f or ev ery i ∈ { 0 , 1 } , consider its pr eimag e A i = ˜ h − 1 1 ( A ) ⊂ C i × Σ i and its pro j ecti on pr i ( A i ) onto the C an tor set C i . It follo ws from A ⊂ K and T 1 i =0 ˜ h i ( C i × Σ i ) = ∅ that eac h s et A i is an analytic subspace of the coanalytic set K i . S in ce the space K i is not analytic, there is a p oint c i ∈ K i \ pr i ( A i ). It follo ws that 1 [ i =0 ˜ h i ( { c i } × Σ i ) = ( a − 1 + c 0 + Σ 0 ) ∪ ( − c 1 − Σ 1 ) ⊂ K \ A and hence ( K \ A ) − ( K \ A ) ⊃ a − 1 + c 0 + Σ 0 + c 1 + Σ 1 ⊃ a − 1 + c 0 + c 1 + B (2 ε − 1 ) ⊃ a − 1 + B ( ε − 1 ) = U according to Claim 1(2). The inclusion B ( ε − 1 ) ⊂ c 0 + c 1 + B (2 ε − 1 ) follo ws from c 0 + c 1 ∈ C 0 + C 1 ⊂ B ( ε 0 ) + B ( ε 0 ) ⊂ B (2 ε 0 ) ⊂ B ( ε − 1 ).  6 T ARAS BANAKH AND NADY A L Y ASKO VSKA Referen ces [1] T. Banakh , N. Lyask o v sk a, We akly P-smal l not P-smal l subsets i n Ab el ian gr oups , Algebra an d Discr. Math. N.3 (2006), 29–34. [2] T. 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Zakrzewski, Uni vers al ly me ager sets , Pro c. Amer. Math. Soc. 129 :6 (2001), 1793–1798. [11] P . Zakrzewski, On a c onstruction of universal ly smal l sets , Real Anal. Exchange 28 :1 (2002/03), 221–226. Instytut ma tema tyki, Un iwersytet Jana Kochano wskiego, Kielce, Poland Dep ar tme nt of Ma thema tics, Iv an Franko Na tional Universi ty of L viv, Ukraine E-mail addr ess : t.o.banakh@ gmail.com, lyaskovska@yahoo .com