Let $p_1,...,p_k$ be a rationally independent sequence of integer valued nonlinear polynomials. We show that for all $E\subseteq \mathbb{N}$, every Folner sequence $\Phi$, and every $\varepsilon>0$, the set $$\left\{n\in \mathbb{N} : d_{\Phi}\left(E\cap(E+p_1(n))\cap\cdots\cap (E+p_k(n))\right)>d_{\Phi}(E)^{k+1}-\varepsilon\right\}$$ intersects every IP generated by a sequence with rational spectrum. Our methods involve the study of the characteristic factors for multiple ergodic polynomial averages along IPs. In particular, we also prove a pointwise convergence theorem for polynomial averages along IPs with rational spectrum, generalizing a well known result of Leibman.
Let $\delta\in\mathbb{F}_{2^n}$ satisfy $\operatorname{Tr}_{\mathbb{F}_{2^n}/\mathbb{F}_2}(\delta)=1$. We study the permutation behavior of $$ f(x) = \left(\frac{1}{x^2+x+\delta}\right)^{2^k}+x $$ over $\mathbb{F}_{2^n}$. Helleseth and Zinoviev proved that $f(x)$ is a permutation for $k=0,1$, and remarked that numerica...
Let $d_1,\ldots,d_r$ be pairwise relatively prime positive square-free integers, with $d_j\geq2$ for all $1\leq j\leq r$. Using elementary matrices and combinatorial mathematics, we give a complete classification of the irreducible $\mathbb{Z}_+$-modules over the domain $\mathbb{Z}[d^{\frac{1}{N_1}}_1,d^{\frac{1}{N_2}}...
Let $\Omega_n$ denote the set of $n\times n$ doubly stochastic matrices. Kim and Roush conjectured in 1981 that, for $n=2k+1>1$, $ \max_{A\in\Omega_{2k+1}}\operatorname{per}(I-A)=3\cdot 2^{k-2}$. They proposed the block construction $A_\star=\frac12(J_3-I_3)\oplus P_2^{\oplus(k-1)}$, where $P_2=\begin{pmatrix}0&1\\1&0\...
Let $\mu$ be the M\"obius function and $e(t)=e^{2\pi it}$. We prove that if $N\ge2$, $\alpha\in\mathbb{R}$, $(a,q)=1$, and $|\alpha-a/q|\le q^{-2}$, then \[\bigg|\sum_{n\le N}\mu^2(n)e(\alpha n)\bigg|\ll\left(\frac Nq+q\right)(\log 2N)^5, \] with an absolute implied constant, and we deduce the corresponding estimate on...
Nicolas Robles, Alexandru Zaharescu, Dirk Zeindler· 0 citations
In this paper, we construct new families of imaginary and real quadratic fields that are $p$-rational. In the imaginary case, we prove that for any positive integer $k$ and any integer $m$, the imaginary quadratic field $\mathbb{Q}\left(\sqrt{-(kp+m)}\right)$ is $p$-rational for sufficiently large primes $p$. The proof...
Let $\mu$ be a log-concave probability measure on $\mathbb R^n$ and let $f\colon\mathbb R^n\to\mathbb R^k$ be a polynomial mapping of degree at most $d$. We show that \[ \mu(f\in A) \le C\bigl(\lambda_k(A)\bigr)^{\frac{1}{k(d-1)+1}} \] for every Borel set $A\subset\mathbb R^k$ whenever the image measure $\mu\circ f^{-1...
Egor D. Kosov, A. Zhukova· 1 citation· ⚡1
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