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In this paper, we consider the classic fair division problem of allocating m divisible items to n agents with linear valuations over the items. We define novel notions of fair shares from the perspective of individual agents via the cake-cutting process. These shares generalize the notion of proportionality by taking into account the valuations of other agents via constraints capturing envy. We study what fraction (approximation) of these shares are achievable in the worst case, and present tight and non-trivial approximation bounds as a function of n and m. In particular, we show a tight approximation bound of Θ(√n) for various notions of such shares. We show this bound via a novel application of dual fitting, which may be of independent interest. We also present a bound of O(m^(2/3)) for a strict notion of share, with an almost matching lower bound. We further develop weaker notions of shares whose approximation bounds interpolate smoothly between proportionality and the shares described above. We finally present empirical results showing that our definitions lead to more reasonable shares than the standard fair share notion of proportionality. 

High-performance optoelectronic devices, such as solar cells and light-emitting diodes, have been fabricated with lead halide perovskites owing to their superior carrier properties. However, charge transport in such optoelectronics is intrinsically directional due to the existence of p–n junctions, which thus lacks the potential to elucidate any perturbations in light or electricity during energy conversion. Here, with the presence of a LiCl additive in a formamidinium chloride (FACl) solution, the as-grown LiCl:FAPbCl 3 nanorods demonstrate greatly enhanced crystallinity and UV photoresponse as compared to pristine FAPbCl 3 nanostructures without the LiCl additive. Most importantly, the LiCl:FAPbCl 3 nanorod film exhibits unprecedented distinguishability to UV photons with different energies and oscillating intensities, in the form of bipolar and periodically oscillatory photocurrents. This work could advance the fundamental understanding of photoinduced carrier processes in halide perovskites and facilitate the development of novel UV-based optical communications.