On the fast Khintchine spectrum in continued fractions

For $x\in [0,1)$, let $x=[a_1(x), a_2(x),...]$ be its continued fraction expansion with partial quotients ${a_n(x), n\ge 1}$. Let $\psi : \mathbb{N} \rightarrow \mathbb{N}$ be a function with $\psi(n)/n\to \infty$ as $n\to \infty$. In this note, the fast Khintchine spectrum, i.e., the Hausdorff dimension of the set E(\psi):=\Big{x\in [0,1): \lim_{n\to\infty}\frac{1}{\psi(n)}\sum_{j=1}^n\log a_j(x)=1\Big} is completely determined without any extra condition on $\psi$.Comment: 10 page

Hausdorff dimension of certain sets arising in continued fraction expansions

AbstractThis paper is concerned with the fractional dimensions of some sets of points with their partial quotients obeying some restrictions in their continued fraction expansions. The Hausdorff dimension of the following set, which shares a dichotomy law according to Borel–Bernstein's theorem, is completely determinedE(ϕ)={x∈[0,1):an(x)⩾ϕ(n), i.o. n}. Our result significantly strengthens the results of Good and Lúczak

A Cross-Category Analysis of the Effects of Consumer Reviews and Ratings

Online consumer ratings and reviews have become very popular among online retailers and are found to influence the sales of experience goods. However, their effects on the sales of search or quasi-experience goods remain unknown. Using data collected from Amazon.com, we examine how the impacts of consumer ratings and reviews differ across product categories. Our results suggest that the sales of experience goods such as books are affected by consumer ratings and reviews, but the sales of quasi-experience and search goods such as digital cameras and USB flash drives are not. In addition, as the price of an experience item increases, consumer ratings and reviews become more influential on sales

A 7.0\% Determination of the Baryon Fraction in the Intergalactic Medium from Localized Fast Radio Bursts

The dispersion measure (DM)--redshift relation of fast radio bursts (FRBs) has been proposed as a potential new tool for probing intergalactic medium (IGM) and for studying cosmology. However, the poor knowledge of the baryon fraction in the IGM ($f_{\mathrm{IGM}}$) and its degeneracy with cosmological parameters impose restrictions on the cosmological applications of FRBs. Furthermore, DMs contributed by the IGM ($\mathrm{DM_{IGM}}$) and host galaxy ($\mathrm{DM_{host}}$), the important cosmological quantities, cannot be exactly extracted from observations, which would bring uncontrolled systematic uncertainties in FRB cosmology. In this work, we use seventeen localized FRBs to constrain $f_{\mathrm{IGM}}$ and possibly its redshift evolution. Other cosmological probes such as Type Ia supernovae, baryon acoustic oscillations, and cosmic microwave background radiation are combined to break parameter degeneracy. Taking into account the probability distributions of $\mathrm{DM_{IGM}}$ and $\mathrm{DM_{host}}$ derived from the the IllustrisTNG simulation, we obtain a robust measurement of $f_{\mathrm{IGM}}=0.857\pm0.060$, representing a precision of 7.0\%. We find that there is no strong evidence for the redshift dependence of $f_{\mathrm{IGM}}$ at the current observational data level. The rapid progress in localizing FRBs will significantly improve the constraints on $f_{\mathrm{IGM}}$.Comment: 9 pages, 3 figures, 2 table

Impact of Consumer Reviews and Ratings on Sales, Prices, and Profits: Theory and Evidence

We provide a theoretical framework to understand the impacts of consumer reviews and ratings on firms’ prices, sales, profits, and consumer surplus. We show how the economic impacts may differ depending on the informativeness of the reviews, the quantifiability of the product attributes, and the competitive environment. First, even though a monopolist always benefits from an improvement in product rating, a firm in a competitive market can hurt by it. When the low quality firm’s product rating improves, its equilibrium profit will decrease if its quality is above a threshold, and increase if its quality is below that threshold. Second, if the high quality firm’s product rating improves, both the high quality and the low quality firm will benefit. Our empirical findings based on point-and-shoot digital camera and multivitamin data collected from Amazon.com provide strong support for these results