Improved Power Laws for the Favard Length Problem in All Dimensions

Abstract

The Favard length of a compact set in ${\mathbb{R}}^d$ is the average one-dimensional Hausdorff measure of its orthogonal projections onto lines. We establish quantitative upper bounds for small neighbourhoods of a class of purely $1$-unrectifiable rational product Cantor sets in every dimension $d\ge 2$. The proof combines coordinatewise small-value estimates, multidimensional sets of large values, and an improved symbolic-cylinder combinatorial argument whose low-multiplicity estimate closes for every parameter $\rho >2$. Applied to the classical four-corner Cantor set, the original Nazarov--Peres--Volberg Fourier input and the improved propagation yield the bound $N^{-1/5+u}$ for every $u>0$. When the coordinate mask polynomials are zero-free on the unit circle, we obtain the general exponent $(2D+1{)}^{-1}-u$, where $D:={\sum }_i{\log }_L(\#A_i/{\min }_{|z|=1}|A_i(z)|)$; in particular, explicit planar examples in bases $25$ and $36$ attain the exponents $1/3-u$ and ${\delta }_{36}-u$, respectively, where ${\delta }_{36}=1/(2{\log }_6(300/53)+1)=0.340720\dots >1/3$.

arXiv comment96 pages, 5 figures, 1 table. Version 2 is substantially revised and expanded: explicit exponent bounds and examples added in Section, strengthened arguments through Sections 3-7, and updated references. Comments welcome!