The set of badly approximable vectors is strongly C1C^1 incompressible

We prove that the countable intersection of $C^1$-diffeomorphic images of certain Diophantine sets has full Hausdorff dimension. For example, we show this for the set of badly approximable vectors in $\mathbb{R}^d$, improving earlier results of Schmidt and Dani. To prove this, inspired by ideas of McMullen, we define a new variant of Schmidt's $(\alpha, \beta)$-game and show that our sets are hyperplane absolute winning (HAW), which in particular implies winning in the original game. The HAW property passes automatically to games played on certain fractals, thus our sets intersect a large class of fractals in a set of positive dimension. This extends earlier results of Fishman to a more general set-up, with simpler proofs.Comment: 18 page

Seeing the vibrational breathing of a single molecule through time-resolved coherent anti-Stokes Raman scattering

The motion of chemical bonds within molecules can be observed in real time, in the form of vibrational wavepackets prepared and interrogated through ultrafast nonlinear spectroscopy. Such nonlinear optical measurements are commonly performed on large ensembles of molecules, and as such, are limited to the extent that ensemble coherence can be maintained. Here, we describe vibrational wavepacket motion on single molecules, recorded through time-resolved, surface-enhanced, coherent anti-Stokes Raman scattering. The required sensitivity to detect the motion of a single molecule, under ambient conditions, is achieved by equipping the molecule with a dipolar nano-antenna (a gold dumbbell). In contrast with measurements in ensembles, the vibrational coherence on a single molecule does not dephase. It develops phase fluctuations with characteristic statistics. We present the time evolution of discretely sampled statistical states, and highlight the unique information content in the characteristic, early-time probability distribution function of the signal.Comment: 17 pages, 5 figure

Ultraviolet and yellow reflectance but not fluorescence is important for visual discrimination of conspecifics by Heliconius erato

Toxic Heliconius butterflies have yellow hindwing bars that – unlike those of their closest relatives – reflect ultraviolet (UV) and long wavelength light, and also fluoresce. The pigment in the yellow scales is 3-hydroxy-DL-kynurenine (3-OHK), which is found in the hair and scales of a variety of animals. In other butterflies like pierids with color schemes characterized by independent sources of variation in UV and human-visible yellow/orange, behavioral experiments have generally implicated the UV component as most relevant to mate choice. This has not been addressed in Heliconius butterflies, where variation exists in analogous color components, but moreover where fluorescence due to 3-OHK could also contribute to yellow wing coloration. In addition, the potential cost due to predator visibility is largely unknown for the analogous well-studied pierid butterfly species. In field studies with butterfly paper models, we show that both UV and 3-OHK yellow act as signals for H. erato when compared with models lacking UV or resembling ancestral Eueides yellow, respectively, but attack rates by birds do not differ significantly between the models. Furthermore, measurement of the quantum yield and reflectance spectra of 3-OHK indicates that fluorescence does not contribute to the visual signal under broad-spectrum illumination. Our results suggest that the use of 3-OHK pigmentation instead of ancestral yellow was driven by sexual selection rather than predation