Let $P (Y_1, ..., Y_d)$ be a certain fixed homogeneous polynomial of integral coefficients. In this paper, we establish a quantitative equidistribution criterion for the ergodic averages along $\Omega (|P (n_1, ..., n_d)|)$. Consequently, by an estimate of Lachand, we prove the following variant of a theorem of Bergelson and Richter: if $P$ is an irreducible binary cubic form and $ (X, T)$ is a uniquely ergodic system with unique invariant measure $\mu$, then for any $x \in X$ and $f \in C(X)$, \begin{equation*} \lim_{N \rightarrow \infty} \frac 1 {N^2} {\mathop{\sum\sum}_{n_1, n_2 \leqslant N}} f \big( T^{ \Omega (|P (n_1, n_2)| ) } x \big) = \int_{X} f \ \mathrm{d} \mu . \end{equation*} Moreover, we prove in the appendix a related conjecture of C\'espedes and Donoso over number fields.
We show that for every fixed $\varepsilon>0$, there exist arbitrarily large families of point-line pairs $(x_1,\ell_1),\ldots,(x_n,\ell_n)$ in $[0,1]^2$, with $x_i \in \ell_i$ for all $i$, and such that $\operatorname{dist}(x_i,\ell_j)\ge n^{-2/3-\varepsilon}$ for all $i \neq j$. Combined with a previous result of Cohen, the author, and Zakharov, this solves the minimal distance problem. The same construction also comes with an unexpected finite field consequence: for every $\varepsilon>0$, there exists a set of primes $q$ of positive relative density for which $\mathbb F_q^2$ contains an induced point-line matching of size $\gtrsim q^{3/2-\varepsilon}$. This disproves a conjecture of Hunter, the author, Verstra\"ete and Zhang.
Let $\varphi_{d}(|{\,\cdot\,}|)$ be the radial extremal function in the problem for the sharp Nikolskii constant $\mathcal C_d^{-1}=\inf \|f\|_{1}$ over functions $f\in\mathit{PW}\,_{1}^{1}(\mathbb R^{d})$, $f(0)=1$. For every odd dimension, we construct an entire function $\Phi$ of exponential type $1/2$ such that $\varphi_d(z)=\Phi(z)\Phi(-z)$, and $\Phi$ satisfies a quadratic functional equation and a second-order linear differential equation with polynomial coefficients. Thus, the problem of finding the extremal function is reduced to a spectral problem with finitely many parameters. This result extends a recent one-dimensional result, but uses a different method. For example, in dimension $d=3$ it leads to a seven-diagonal spectral scheme that allows us to compute $\mathcal C_3$ to high accuracy. Even dimensions remain open within this approach.
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 $\mu\circ f^{-1}$ is absolutely continuous. The constant $C$ is independent of the dimension $n$, and the exponent $\frac{1}{k(d-1)+1}$ is sharp. This extends the scalar Carbery--Wright inequality and answers, in the log-concave setting, a question raised by Avni, Glazer, and Larsen. In addition, we show that the density of $\mu\circ f^{-1}$, whenever it exists, belongs to the Nikolskii--Besov space $B^{\frac{1}{k(d-1)+1}}_{1,\infty}(\mathbb R^k)$, with a dimension-free bound for the corresponding norm. A central difficulty in passing from scalar polynomials to vector-valued polynomial mappings is the lack of a suitable nondegeneracy parameter quantifying absolute continuity of $\mu\circ f^{-1}$, as the variance does in the scalar case. Natural candidates such as the covariance matrix or the Jacobian matrix either fail to characterize this property or do not lead to dimension-free estimates. We identify such a parameter and define it to be the covariance matrix of the vector formed by the monomials of degree up to $d^{k-1}$ in the normalized components of $f$. The dimension-free nature of our results allows us to extend Kusuoka's absolute continuity criterion for Gaussian polynomial random vectors to the log-concave setting. Moreover, in this setting, we obtain estimates relating convergence in distribution to convergence in total variation for polynomial random vectors.
In this paper, we refine Borel's 1903 result in Diophantine approximation by providing sharper bounds for the minimum of three consecutive approximation coefficients $\Theta_n(x)$, defined for any real number $x$ with regular continued fraction (RCF) expansion $x=[0;a_1,a_2,\dots]$ as $\Theta_n = q_n^2\left| x-\frac{p_n}{q_n}\right|$. Here $\frac{p_n}{q_n}$ is the $n$th RCF convergent of $x$. Borel's result states that for all (irrational) $x$ and all $n\in\mathbb{N}$, $$ \min \left\{ \Theta_{n-1}(x),\Theta_n(x),\Theta_{n+1}(x)\right\} \leq \frac{1}{\sqrt{5}}. $$ We focus on the situation where $a_{n+1}=1$, since otherwise a result by F.~Bagemihl and J.R.~McLaughlin from 1966 implies that the Borel-bound $1/\sqrt{5}$ can already be improver to $1/\sqrt{8}$.
Savita S. Adhin, Ayreena Bakhtawar, C. Kraaikamp· 0 citations
Let $\Omega(n)$ denote the number of prime factors of $n$, counted with multiplicities. We study the set $Acc^\Omega(x)$ of weak-$^*$ limits of the sequence $\frac{1}{N}\sum_{n\leq N}\delta_{T^{\Omega(n)}x}$ in $\sigma$-compact dynamical systems $ (X,T)$, demonstrating that if $x \in X$ is quasi-generic for an ergodic measure $\mu$, then $\mu \in Acc^\Omega(x)$. This extends a result of Bergelson and Richter, who studied the problem in the setting of uniquely ergodic systems. We give a more precise description of the set $Acc^\Omega(x)$ in the case of the horocycle flow on non-compact quotients of $SL(2,\mathbb{R})$. We show that for every non-periodic $x\in X$, in addition to Haar measure, there exists sequences $(s_n), (c_n) \subseteq \mathbb{R}$ such that $$ \frac{1}{\sqrt{2\pi}}\int_{-\infty}^{\infty}e^{-\frac{r^2}{2}}\nu^{i}_{s_n-2\log|1+c_nr|} dr\in Acc^{\Omega}(x), $$ where $\{ \nu^{i}_{s} \}_{i \leq k}$ denotes the one parameter family of periodic measures in each of the $k$ inequivalent cusps. Depending on Diophantine properties of the non-periodic point $x$, we show that $Acc^\Omega(x)$ contains a full two parameter family of such periodic measures, as well as the Dirac measure at each cusp. In particular, these results yield almost-everywhere divergence of pointwise averages along $\Omega(n)$ for the non-compact horocycle flow.
Let $\Omega(n)$ denote the number of prime factors of $n$, counted with multiplicity. We prove that if $A\subset\mathbb{N}^3$ has positive upper Banach density, then there are $(x,y,z)\in\mathbb{N}^3$ and $d\in\mathbb{N}$ such that $$(x,y,z),(x+d,y,z),(x,y+d,z),(x,y,z+\Omega(d))\in A.$$ To establish the above result, we give an $L^2$-decoupling theorem for the triple ergodic averages $$ \frac1N\sum_{n=1}^N T_1^n f_1\,T_2^n f_2\,S^{\Omega(n)}g $$ associated with three commuting transformations by isotropy factors and nilpotent structures in $\mathbb{Z}^2$-actions.
