scholarly journals Integral Menger curvature and rectifiability of $n$-dimensional Borel sets in Euclidean $N$-space

2017 ◽  
Vol 370 (2) ◽  
pp. 1185-1250 ◽  
Author(s):  
Martin Meurer
Keyword(s):  
Borel Sets ◽  
Menger Curvature

ERRATA TO "ON BOREL SETS WITH SMALL COVER"

1993 ◽  
Vol 19 (1) ◽  
pp. 58 ◽  
Author(s):  
Petruska
Keyword(s):  
Borel Sets ◽  
Small Cover

scholarly journals Decomposing the real line into Borel sets closed under addition

2015 ◽  
Vol 61 (6) ◽  
pp. 466-473
Author(s):  
Márton Elekes ◽  
Tamás Keleti
Keyword(s):  
Real Line ◽  
Borel Sets ◽  
The Real

scholarly journals Homogeneous Borel sets

1986 ◽  
Vol 96 (4) ◽  
pp. 673-673 ◽  
Author(s):  
Fons van Engelen
Keyword(s):  
Borel Sets

scholarly journals Dimensions of Fractional Brownian Images

2021 ◽  
Author(s):  
Stuart A. Burrell
Keyword(s):  
Brownian Motion ◽  
Hausdorff Dimension ◽  
Stochastic Processes ◽  
Fractional Brownian Motion ◽  
Box Dimension ◽  
General Strategy ◽  
Borel Sets ◽  
Exceptional Sets ◽  
Explicit Condition ◽  
Box Dimensions

AbstractThis paper concerns the intermediate dimensions, a spectrum of dimensions that interpolate between the Hausdorff and box dimensions. Potential-theoretic methods are used to produce dimension bounds for images of sets under Hölder maps and certain stochastic processes. We apply this to compute the almost-sure value of the dimension of Borel sets under index-$$\alpha $$ α fractional Brownian motion in terms of dimension profiles defined using capacities. As a corollary, this establishes continuity of the profiles for Borel sets and allows us to obtain an explicit condition showing how the Hausdorff dimension of a set may influence the typical box dimension of Hölder images such as projections. The methods used propose a general strategy for related problems; dimensional information about a set may be learned from analysing particular fractional Brownian images of that set. To conclude, we obtain bounds on the Hausdorff dimension of exceptional sets, with respect to intermediate dimensions, in the setting of projections.

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