Productinequalities Packingmeasures Productformulas Thinsets Dimensionofpowers Examplesandproblems Packing dimensions and cartesian products Ondˇrej Zindulka CzechTechnicalUniversityPrague [email protected] St Andrews 2007 OndˇrejZindulka Packingdimensionsandcartesianproducts Productinequalities Packingmeasures Productformulas Thinsets Dimensionofpowers Examplesandproblems Hausdorff measure Lebesgue measure: (cid:110)(cid:88) (cid:111) L(E)=supinf diamI :{I } a cover of E by intervals of length <δ i i δ>0 i s-dimensional Hausdorff measure (s>0): (cid:110)(cid:88) (cid:111) Hs(E)=supinf (diamE )s :{E } is a δ-cover of E i i δ>0 i Hausdorff dimension: dim E =inf{s:HsE =0}=sup{s:HsE =∞} H OndˇrejZindulka Packingdimensionsandcartesianproducts Productinequalities Packingmeasures Productformulas Thinsets Dimensionofpowers Examplesandproblems Packing measure Packing of E ⊆X: Disjoint collection of balls B(x,r) with x∈E. s-dimensional packing pre-measure: (cid:110)(cid:88) (cid:111) Ps(E)= inf sup (2r )s :{B(x ,r )} is a δ-packing of E 0 i i i δ>0 i s-dimensional packing measure: (cid:110)(cid:88) (cid:111) Ps(E)=inf Ps(E ):{E } is a cover of E n n n Packing dimension: dim E =inf{s:Ps(E)=0}=sup{s:Ps(E)=∞}. P Fact Hs(E)(cid:54)Ps(E) dim E (cid:54)dim E H P OndˇrejZindulka Packingdimensionsandcartesianproducts Productinequalities Packingmeasures Productformulas Thinsets Dimensionofpowers Examplesandproblems Cartesian product inequalities X,Y separable metric spaces. Provide X×Y with a maximum metric: Balls are squares. Theorem (Marstrand, Tricot, Howroyd...) dim X+dim Y (cid:54)dim X×Y H H H (cid:54) dim X+dim Y (cid:54)dim X×Y H P P (cid:54)dim X+dim Y P P OndˇrejZindulka Packingdimensionsandcartesianproducts Productinequalities Packingmeasures Productformulas Thinsets Dimensionofpowers Examplesandproblems Hu & Taylor’s definition dim X (cid:54)dim X×Y −dim Y H P P dim X (cid:54)inf {dim X×Y −dim Y} H Y P P Definition (Hu & Taylor ’94) X ⊆Rn : aDimX =inf{dim X×Y −dim Y :Y ⊆Rn Borel} P P We know: dim X (cid:54)aDimX H Question (Hu & Taylor): Is dim X =aDimX? H Answer (Bishop & Peres, Xiao ’96): No! OndˇrejZindulka Packingdimensionsandcartesianproducts Productinequalities Packingmeasures Productformulas Thinsets Dimensionofpowers Examplesandproblems Upper box-counting dimension Box-counting function: N (δ)=min{|A|: A is a δ-cover of E} E Upper box-counting dimension: logN (δ) dim E =limsup E B |logδ| δ→0 Upper packing dimension: dim E = inf supdim E P (cid:83)En=X n B n Theorem (well-known) dim X =dim X P P OndˇrejZindulka Packingdimensionsandcartesianproducts Productinequalities Packingmeasures Productformulas Thinsets Dimensionofpowers Examplesandproblems Lower box-counting dimension Box-counting function: N (δ)=min{|A|: A is a δ-cover of E} E Lower box-counting dimension: logN (δ) dim E =liminf E B δ→0 |logδ| Lower packing dimension: dim E = inf supdim E P (cid:83)En=X n B n Theorem (Tricot ’82) dim X (cid:62)dim X (but not dim X =dim X). P H P H OndˇrejZindulka Packingdimensionsandcartesianproducts Productinequalities Packingmeasures Productformulas Thinsets Dimensionofpowers Examplesandproblems Improving dim X (cid:54) aDimX H Theorem (Bishop & Peres, Xiao ’96) If X ⊆Rn is compact and Y ⊆Rn is Borel, then dim X+dim Y (cid:54)dim X×Y P P P hence dim X (cid:54)aDimX P But not dim X =aDimX. P Limitations of the proofs: B&P: Rather special representation of compact sets in Rn Xiao: Baire category theorem Both about dimension rather than measures OndˇrejZindulka Packingdimensionsandcartesianproducts Productinequalities Packingmeasures Productformulas Thinsets Dimensionofpowers Examplesandproblems Goals Define lower packing measure so that dim X =inf{s:Ps(X)=0}=sup{s:Ps(X)=∞} P Show, for arbitrary metric spaces, Ps(X)·Pt(Y)(cid:54)Ps+t(X×Y) Modify dim so that P dim X (cid:54)aDimX becomes dim X =aDimX P −−→P OndˇrejZindulka Packingdimensionsandcartesianproducts Productinequalities Packingmeasures Productformulas Thinsets Dimensionofpowers Examplesandproblems Packing measures revisited Joyce & Preiss ’95, Edgar ’01,’07: Packing revised: {B(x ,r )} s.t. x ∈/ B(x ,r ) i i j i i Hausdorff function: g :[0,∞)→[0,∞) nondecreasing g(r)=0 iff r=0 no continuity required Notation: ∆(cid:32)0 means ∆⊆R, 0∈∆ A packing π ={B(x ,r )} is ∆-valued if r ∈∆ i i i (cid:80) g(π)= g(r ) i i Definition (Lower packing pre-measure) Pg(E)= inf sup{g(π):π is a ∆-valued packing of E} 0 ∆(cid:32)0 OndˇrejZindulka Packingdimensionsandcartesianproducts