Structure of multicorrelation sequences with integer part polynomial iterates along primes

Let $T$ be a measure preserving $\mathbb{Z}^\ell$-action on the probability space $(X,{\mathcal B},\mu),$ $q_1,\dots,q_m:{\mathbb R}\to{\mathbb R}^\ell$ vector polynomials, and $f_0,\dots,f_m\in L^\infty(X)$. For any $\epsilon > 0$ and multicorrelation sequences of the form $\displaystyle\alpha(n)=\int_Xf_0\cdot T^{ \lfloor q_1(n) \rfloor }f_1\cdots T^{ \lfloor q_m(n) \rfloor }f_m\;d\mu$ we show that there exists a nilsequence $\psi$ for which $\displaystyle\lim_{N - M \to \infty} \frac{1}{N-M} \sum_{n=M}^{N-1} |\alpha(n) - \psi(n)| \leq \epsilon$ and $\displaystyle\lim_{N \to \infty} \frac{1}{\pi(N)} \sum_{p \in {\mathbb P}\cap[1,N]} |\alpha(p) - \psi(p)| \leq \epsilon.$ This result simultaneously generalizes previous results of Frantzikinakis [2] and the authors [11,13].Comment: 7 page

Medium Resolution Near-Infrared Spectra of the Host Galaxies of Nearby Quasars

We present medium resolution near-infrared host galaxy spectra of low redshift quasars, PG 0844 + 349 (z=0.064), PG 1226 + 023 (z=0.158), and PG 1426+015 (z=0.086). The observations were done by using the Infrared Camera and Spectrograph (IRCS) at the Subaru 8.2 m telescope. The full width at half maximum of the point spread function was about 0.3 arcsec by operations of an adaptive optics system, which can effectively resolve the quasar spectra from the host galaxy spectra. We spent up to several hours per target and developed data reduction methods to reduce the systematic noises of the telluric emissions and absorptions. From the obtained spectra, we identified absorption features of Mg I (1.503 um), Si I (1.589 um) and CO (6-3) (1.619 um), and measured the velocity dispersions of PG 0844 + 349 to be 132+/-110 km s-1 and PG 1426 + 015 to be 264+/-215 km s-1. By using an M_BH-sigma relation of elliptical galaxies, we derived the black hole (BH) mass of PG 0844+349, log(M_BH/M_SUN) = 7.7+/-5.5 and PG 1426+015, log(M_BH/M_SUN) = 9.0+/-7.5. These values are consistent with the BH mass values from broad emission lines with an assumption of a virial factor of 5.5.Comment: 16 pages, 5 figure