Statement of results Proof outline for measure Small unions of affine subspaces and skeletons via Baire category Alan Chang (ac@math.uchicago.edu) http://math.uchicago.edu/~ac Joint work with Marianna Csörnyei, Kornélia Héra and Tamás Keleti Workshop on Geometric Measure Theory 10-14 July 2017 The University of Warwick, Coventry, UK 1
Statement of results Proof outline for measure Alan Chang, Marianna Csörnyei, Kornélia Héra and Tamás Keleti 2
Statement of results Proof outline for measure Previous results Theorem (Stein 1976) Let n ≥ 3 . Any subset of R n which contains an ( n − 1 ) -sphere centred at each point of R n must have positive Lebesgue measure. 3
Statement of results Proof outline for measure Previous results Theorem (Stein 1976) Let n ≥ 3 . Any subset of R n which contains an ( n − 1 ) -sphere centred at each point of R n must have positive Lebesgue measure. Theorem (Bourgain 1986, Marstrand 1987) Same holds for n = 2 . 4
Statement of results Proof outline for measure Previous results Theorem (Stein 1976) Let n ≥ 3 . Any subset of R n which contains an ( n − 1 ) -sphere centred at each point of R n must have positive Lebesgue measure. Theorem (Bourgain 1986, Marstrand 1987) Same holds for n = 2 . What if we use something other than circles? 5
Statement of results Proof outline for measure Previous results Theorem (Stein 1976) Let n ≥ 3 . Any subset of R n which contains an ( n − 1 ) -sphere centred at each point of R n must have positive Lebesgue measure. Theorem (Bourgain 1986, Marstrand 1987) Same holds for n = 2 . What if we use something other than circles? Theorem (Keleti, Nagy, and Shmerkin 2014) In R 2 , there exists a closed set of Hausdorff dimension 1 (and hence Lebesgue measure zero) that contains the boundary of an axis-parallel square around each point of R 2 . 6
Statement of results Proof outline for measure The kinds of questions we ask Let S ⊂ R n be the k -skeleton of a polytope. 7
Statement of results Proof outline for measure The kinds of questions we ask Let S ⊂ R n be the k -skeleton of a polytope. Suppose we “put a copy” of S around every point of R n . What is the minimal dimension/measure of the resulting set? 8
Statement of results Proof outline for measure The kinds of questions we ask Let S ⊂ R n be the k -skeleton of a polytope. Suppose we “put a copy” of S around every point of R n . What is the minimal dimension/measure of the resulting set? Wait... what do we mean by “copy”? 9
Statement of results Proof outline for measure The kinds of questions we ask Let S ⊂ R n be the k -skeleton of a polytope. Suppose we “put a copy” of S around every point of R n . What is the minimal dimension/measure of the resulting set? Wait... what do we mean by “copy”? a scaled copy: x + rS ( r > 0) a rotated copy: x + T ( S ) ( T ∈ SO ( n ) ) a scaled and rotated copy: x + rT ( S ) 10
Statement of results Proof outline for measure (Corollaries of) our results Theorem (C., Csörnyei, Héra, Keleti) For any integers 0 ≤ k < n, the minimal dimension of a Borel set A ⊂ R n that contains the k-skeleton of 1. a scaled copy of a cube around every point of R n is n − 1 [Thornton 2015]; 11
Statement of results Proof outline for measure (Corollaries of) our results Theorem (C., Csörnyei, Héra, Keleti) For any integers 0 ≤ k < n, the minimal dimension of a Borel set A ⊂ R n that contains the k-skeleton of 1. a scaled copy of a cube around every point of R n is n − 1 [Thornton 2015]; 2. a scaled and rotated copy of a cube around every point of R n is k; 12
Statement of results Proof outline for measure (Corollaries of) our results Theorem (C., Csörnyei, Héra, Keleti) For any integers 0 ≤ k < n, the minimal dimension of a Borel set A ⊂ R n that contains the k-skeleton of 1. a scaled copy of a cube around every point of R n is n − 1 [Thornton 2015]; 2. a scaled and rotated copy of a cube around every point of R n is k; 3. a rotated copy of a cube around every point of R n is k + 1 ; 13
Statement of results Proof outline for measure (Corollaries of) our results Theorem (C., Csörnyei, Héra, Keleti) For any integers 0 ≤ k < n, the minimal dimension of a Borel set A ⊂ R n that contains the k-skeleton of 1. a scaled copy of a cube around every point of R n is n − 1 [Thornton 2015]; 2. a scaled and rotated copy of a cube around every point of R n is k; 3. a rotated copy of a cube around every point of R n is k + 1 ; 4. a rotated cube of every size around every point of R n is k + 1 . 14
Statement of results Proof outline for measure (Corollaries of) our results Theorem (C., Csörnyei, Héra, Keleti) For any integers 0 ≤ k < n, the minimal dimension of a Borel set A ⊂ R n that contains the k-skeleton of 1. a scaled copy of a cube around every point of R n is n − 1 [Thornton 2015]; 2. a scaled and rotated copy of a cube around every point of R n is k; 3. a rotated copy of a cube around every point of R n is k + 1 ; 4. a rotated cube of every size around every point of R n is k + 1 . Theorem (C., Csörnyei, Héra, Keleti) Let n ≥ 2 . Then there exists a set of Lebesgue measure zero that contains the boundary (i.e., ( n − 1 ) -skeleton) of a rotated cube of every size around every point of R n . 15
Statement of results Proof outline for measure Strategy (Baire Category) To prove these results, we show that in these cases, the minimal dimension/measure is also the typical one. (We also found situations where minimal � = typical.) Recall that “typical” means “on a residual set” 16
Statement of results Proof outline for measure Strategy (Baire Category) To prove these results, we show that in these cases, the minimal dimension/measure is also the typical one. (We also found situations where minimal � = typical.) Recall that “typical” means “on a residual set” General strategy: Prove residuality for a single affine k -plane not containing 0. Use Baire Category Theorem to obtain result for countable unions. 17
Statement of results Proof outline for measure Statement of result For simplicity we work in R 2 . Let P 1 denote the set of all directions in R 2 . (We identify it with R /π Z .) 18
Statement of results Proof outline for measure Statement of result For simplicity we work in R 2 . Let P 1 denote the set of all directions in R 2 . (We identify it with R /π Z .) For ( x , θ ) ∈ R 2 × P 1 , we let ℓ ( x , θ ) ⊂ R 2 be the line through x in direction θ (i.e., making angle θ with the horizontal axis). 19
Statement of results Proof outline for measure Statement of result For simplicity we work in R 2 . Let P 1 denote the set of all directions in R 2 . (We identify it with R /π Z .) For ( x , θ ) ∈ R 2 × P 1 , we let ℓ ( x , θ ) ⊂ R 2 be the line through x in direction θ (i.e., making angle θ with the horizontal axis). Fix a nonempty compact set C ⊂ R 2 . Let P = { K ⊂ C × P 1 : K compact, has full projection onto C } . 20
Statement of results Proof outline for measure Statement of result For simplicity we work in R 2 . Let P 1 denote the set of all directions in R 2 . (We identify it with R /π Z .) For ( x , θ ) ∈ R 2 × P 1 , we let ℓ ( x , θ ) ⊂ R 2 be the line through x in direction θ (i.e., making angle θ with the horizontal axis). Fix a nonempty compact set C ⊂ R 2 . Let P = { K ⊂ C × P 1 : K compact, has full projection onto C } . Fact: P is a complete metric space with respect to the Hausdorff metric d ( A , B ) = inf { ǫ ≥ 0 : A ⊂ B ǫ and B ⊂ A ǫ } . 21
Statement of results Proof outline for measure Statement of result P = { K ⊂ C × P 1 : K compact, has full projection onto C } � A K = ℓ ( x , θ ) ( x ,θ ) ∈ K 22
Statement of results Proof outline for measure Statement of result P = { K ⊂ C × P 1 : K compact, has full projection onto C } � A K = ℓ ( x , θ ) ( x ,θ ) ∈ K We’ll sketch the proof of the following Nikodym-type result. Theorem (C., Csörnyei, Héra, Keleti) For a typical K ∈ P , the set � ℓ ( x , θ ) \ { x } ( x ,θ ) ∈ K has measure zero. 23
Statement of results Proof outline for measure Statement of result P = { K ⊂ C × P 1 : K compact, has full projection onto C } � A K = ℓ ( x , θ ) ( x ,θ ) ∈ K We’ll sketch the proof of the following Nikodym-type result. Theorem (C., Csörnyei, Héra, Keleti) For a typical K ∈ P , the set � ℓ ( x , θ ) \ { x } ( x ,θ ) ∈ K has measure zero. (The same ideas are used in the previously stated theorem about boundaries of cubes.) 24
Statement of results Proof outline for measure Turning it into a geometric problem P = { K ⊂ C × P 1 : K compact, has full projection onto C } � A K = ℓ ( x , θ ) ( x ,θ ) ∈ K 25
Statement of results Proof outline for measure Turning it into a geometric problem P = { K ⊂ C × P 1 : K compact, has full projection onto C } � A K = ℓ ( x , θ ) ( x ,θ ) ∈ K Fix a ball B ⊂ R 2 . Given any K ∈ P , can we find an L ∈ P close to K , such that A L ∩ B is small? ( B is “bad.”) 26
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