@prefix rdf: .
@prefix rdfs: .
@prefix owl: .
@prefix xsd: .
@prefix nif: .
@prefix itsrdf: .
a nif:String , nif:Context , nif:RFC5147String ;
nif:isString """I’ve just uploaded a new paper to the arXiv entitled “ A quantitative form of the Besicovitch projection theorem via multiscale analysis “, submitted to the Journal of the London Mathematical Society . In the spirit of my earlier posts on soft and hard analysis, this paper establishes a quantitative version of a well-known theorem in soft analysis, in this case the Besicovitch projection theorem. This theorem asserts that if a subset E of the plane has finite length (in the Hausdorff sense ) and is purely unrectifiable (thus its intersection with any Lipschitz graph has zero length), then almost every linear projection E to a line will have zero measure. (In contrast, if E is a rectifiable set of positive length, then it is easy to show that all but at most one linear projection of E will have positive measure, basically thanks to the Rademacher differentiation theorem .)
[...]
A concrete special case of this theorem relates to the product Cantor set K, consisting of all points (x,y) in the unit square whose base 4 expansion consists only of 0s and 3s. This is a compact one-dimensional set of finite length, which is purely unrectifiable, and so Besicovitch’s theorem tells us that almost every projection of K has measure zero. (One consequence of this, first observed by Kahane , is that one can construct Kakeya sets in the plane of zero measure by connecting line segments between one Cantor set and another.)
[...]
I’ve just uploaded a new paper to the arXiv entitled “ A quantitative form of the Besicovitch projection theorem via multiscale analysis “, submitted to the Journal of the London Mathematical Society . In the spirit of my earlier posts on soft and hard analysis, this paper establishes a quantitative version of a well-known theorem in soft analysis, in this case the Besicovitch projection theorem. This theorem asserts that if a subset E of the plane has finite length (in the Hausdorff sense ) and is purely unrectifiable (thus its intersection with any Lipschitz graph has zero length), then almost every linear projection E to a line will have zero measure. (In contrast, if E is a rectifiable set of positive length, then it is easy to show that all but at most one linear projection of E will have positive measure, basically thanks to the Rademacher differentiation theorem .) A concrete special case of this theorem relates to the product Cantor set K, consisting of all points (x,y) in the unit square whose base 4 expansion consists only of 0s and 3s. This is a compact one-dimensional set of finite length, which is purely unrectifiable, and so Besicovitch’s theorem tells us that almost every projection of K has measure zero. (One consequence of this, first observed by Kahane , is that one can construct Kakeya sets in the plane of zero measure by connecting line segments between one Cantor set and another.) Read the rest of this entry »"""^^xsd:string;
nif:beginIndex "0"^^xsd:nonNegativeInteger;
nif:endIndex "2891"^^xsd:nonNegativeInteger;
nif:sourceUrl .
a nif:String , nif:RFC5147String , nif:Snippet ;
nif:referenceContext ;
nif:beginIndex "0"^^xsd:nonNegativeInteger ;
nif:endIndex "884"^^xsd:nonNegativeInteger ;
nif:wasConvertedFrom .
a nif:String , nif:RFC5147String ;
nif:referenceContext ;
nif:anchorOf """Hausdorff sense"""^^xsd:string ;
nif:beginIndex "479"^^xsd:nonNegativeInteger ;
nif:endIndex "494"^^xsd:nonNegativeInteger ;
nif:wasConvertedFrom ;
a nif:Phrase ;
itsrdf:taIdentRef .
a nif:String , nif:RFC5147String , nif:Snippet ;
nif:referenceContext ;
nif:beginIndex "891"^^xsd:nonNegativeInteger ;
nif:endIndex "1430"^^xsd:nonNegativeInteger ;
nif:wasConvertedFrom .
a nif:String , nif:RFC5147String ;
nif:referenceContext ;
nif:anchorOf """Kakeya sets"""^^xsd:string ;
nif:beginIndex "1325"^^xsd:nonNegativeInteger ;
nif:endIndex "1336"^^xsd:nonNegativeInteger ;
nif:wasConvertedFrom ;
a nif:Phrase ;
itsrdf:taIdentRef .
a nif:String , nif:RFC5147String , nif:SnippetSeparator;
nif:referenceContext ;
nif:beginIndex "885"^^xsd:nonNegativeInteger ;
nif:endIndex "891"^^xsd:nonNegativeInteger ;
nif:anchorOf "[...]"^^xsd:string .
a nif:String , nif:RFC5147String , nif:Snippet ;
nif:referenceContext ;
nif:beginIndex "1437"^^xsd:nonNegativeInteger ;
nif:endIndex "2891"^^xsd:nonNegativeInteger ;
nif:wasConvertedFrom .
a nif:String , nif:RFC5147String ;
nif:referenceContext ;
nif:anchorOf """Rademacher differentiation theorem"""^^xsd:string ;
nif:beginIndex "2284"^^xsd:nonNegativeInteger ;
nif:endIndex "2318"^^xsd:nonNegativeInteger ;
nif:wasConvertedFrom ;
a nif:Phrase ;
itsrdf:taIdentRef .
a nif:String , nif:RFC5147String , nif:SnippetSeparator;
nif:referenceContext ;
nif:beginIndex "1431"^^xsd:nonNegativeInteger ;
nif:endIndex "1437"^^xsd:nonNegativeInteger ;
nif:anchorOf "[...]"^^xsd:string .