A new dynamical proof of the Shmerkin–Wu theorem

<p style='text-indent:20px;'>Let <inline-formula><tex-math id="M1">\begin{document}$ a &lt; b $\end{document}</tex-math></inline-formula> be multiplicatively independent integers, both at least <inline-formula><tex-math id="M2">\begin{document}$ 2 $\end{document}</tex-math></inline-formula>. Let <inline-formula><tex-math id="M3">\begin{document}$ A,B $\end{document}</tex-math></inline-formula> be closed subsets of <inline-formula><tex-math id="M4">\begin{document}$ [0,1] $\end{document}</tex-math></inline-formula> that are forward invariant under multiplication by <inline-formula><tex-math id="M5">\begin{document}$ a $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M6">\begin{document}$ b $\end{document}</tex-math></inline-formula> respectively, and let <inline-formula><tex-math id="M7">\begin{document}$ C : = A\times B $\end{document}</tex-math></inline-formula>. An old conjecture of Furstenberg asserted that any planar line <inline-formula><tex-math id="M8">\begin{document}$ L $\end{document}</tex-math></inline-formula> not parallel to either axis must intersect <inline-formula><tex-math id="M9">\begin{document}$ C $\end{document}</tex-math></inline-formula> in Hausdorff dimension at most <inline-formula><tex-math id="M10">\begin{document}$ \max\{\dim C,1\} - 1 $\end{document}</tex-math></inline-formula>. Two recent works by Shmerkin and Wu have given two different proofs of this conjecture. This note provides a third proof. Like Wu's, it stays close to the ergodic theoretic machinery that Furstenberg introduced to study such questions, but it uses less substantial background from ergodic theory. The same method is also used to re-prove a recent result of Yu about certain sequences of sums.</p>

Vertically Integrated Combline Filters in Multilayer PCB

Vertical stacking of planar line resonators in multilayer media is an effective footprint reduction technique for high order filters, but strong broadside line coupling often negates the possibility of narrowband filters. It is shown that vertically stacking combline resonators with rotated centerlines leads to moderate bandwidth filters. Four variations of a 5th order filter are demonstrated, achieving 1.92 – 3.3 dB IL over 10% FBW around 10 GHz while occupying between 6.62 – 22.47 mm2 surface area on a 7 layer PCB.

Vertically Integrated Combline Filters in Multilayer PCB

Vertical stacking of planar line resonators in multilayer media is an effective footprint reduction technique for high order filters, but strong broadside line coupling often negates the possibility of narrowband filters. It is shown that vertically stacking combline resonators with rotated centerlines leads to moderate bandwidth filters. Four variations of a 5th order filter are demonstrated, achieving 1.92 – 3.3 dB IL over 10% FBW around 10 GHz while occupying between 6.62 – 22.47 mm2 surface area on a 7 layer PCB.

Geometric probability, psychophysics and invariance

The perception of the noisy visual world around us naturally combines geometry and probability with psychophysics. So how do we perceive geometric objects from a probabilistic perspective, i.e. infer randomness in a spatial setting ? To test this psychophysically, we use a set of simple experiments to distinguish between probability distributions of planar line images connected with Buffon’s needle and Bertrand’s paradox, two classic exemplars of geometric probability. We find that participants associate greater randomness with images that are invariant under the sub-groups of translation, rotation, and scale. An information theoretic framework centered around the Radon (Hough) transform captures the observed behavioral results, and suggests that symmetry and chance are embedded in human visual perception.