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- W1619071346 abstract "We prove the following variant of Marstrand's theorem about projections of cartesian products of sets: Consider the space $Lambda_m=set{(t,O), tinR, Oin SO(m)}$ with the natural measure and set $Lambda=Lambda_{m_1}timesppptimesLambda_{m_n}$. For every $la=(t_1,O_1,ppp,t_n,O_n)inLambda$ and every $x=(x^1,ppp,x^n)inR^{m_1}timesppptimesR^{m_n}$ we define $pi_la(x)=pi(t_1O_1x^1,ppp,t_nO_nx^n)$. Suppose that $pi$ is surjective and set $$mathfrak{m}:=minset{sum_{iin I}dim_H(K_i) + dimpi(bigoplus_{iin I^c}R^{m_i}), Isubsetset{1,ppp,n}, Ineemptyset}.$$ Then we have {thm*} emph{(i)} If $mathfrak{m}>k$, then $pi_la(K_1timesppptimes K_n)$ has positive $k$-dimensional Lebesgue measure for almost every $lainLambda$. emph{(ii)} If $mathfrak{m}leq k$ and $dim_H(K_1timesppptimes K_n)=dim_H(K_1)+ppp+dim_H(K_n)$, then $dim_H(pi_la(K_1timesppptimes K_n))=mathfrak{m}$ for almost every $lainLambda$. {thm*}" @default.
- W1619071346 created "2016-06-24" @default.
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- W1619071346 date "2011-06-28" @default.
- W1619071346 modified "2023-09-27" @default.
- W1619071346 title "A variant of Marstrand's theorem for projections of cartesian products" @default.
- W1619071346 hasPublicationYear "2011" @default.
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