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 $1/\sqrt{8}$.
For fixed $\tau>0$, non-integer $\theta>2$ and large enough $s$, we investigate the number of solutions to the Diophantine inequality $|x_1^{\theta}+\cdots +x_s^{\theta} - R|<\tau$ as $R \to \infty$. Here, we restrict the variables $x_i$ in the ``almost diagonal"range $X-Y<x_i \leq X+Y$ for $i = 1, \ldots, s$, where $X = (R/s)^{1/\theta}$ and $Y \asymp \sqrt{X}$. Let $\omega = (\lfloor s/2 \rfloor(\theta-1))^{-1/2}$. We will show that if $Y = c\sqrt{X}$ for some $c>\omega$ then for sufficiently large $R$ there must always exist solutions, but if $c<\omega$ then there exist arbitrarily large positive $R$ for which there are no solutions. Our result is thus essentially sharp, with the exception of $c = \omega$. This work is analogous to the results of Daemen and Wright in which similar statements are proved for Waring's problem, though our results are likely somewhat stronger than what is possible in that setting. We discuss other related results and further work to be done.
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.
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.
Let $a(n)$ and $b(n)$ be arithmetic sequences, and $$A(s)=\sum_{n\ge1}a(n)n^{-s}, \qquad B(s)=\sum_{n\ge1}b(n)n^{-s},$$ be the two Dirichlet series related by a certain functional equation. Let $m$ be the \emph{analytic degree} of the functional equation. For $x>0$ and a positive integer $N$, Friedlander and Iwaniec (2005) define the sharply truncated nonlinear dual sum $$\mathcal B_{\ell,D}(x,N) := \sum_{\substack{n\in\mathbb N\\ n\le N}} b(n)n^{-\beta_m} \cos\left( 2\pi m\left(\frac{nx}{D}\right)^{1/m} +\frac{\pi\ell}{4} \right),$$ where $D\ge1$ is the conductor, $\beta_m:=\frac{m+1}{2m}$, and $\ell=m-3-2k$ is determined by the archimedean weight $k$ of the functional equation. Their Conjecture 1 predicts that, for every $\varepsilon>0$, $$\mathcal B_{\ell,D}(x,N) \ll_{\varepsilon,\boldsymbol\kappa} (DNx)^\varepsilon,$$ uniformly in the variables $x$ and $N$, with the degree, conductor, and archimedean datum fixed. We give counterexamples to this prediction with $$A(s)=B(s)=\zeta(s)^m,\; m\geq 4$$ where $$\zeta(s):=\sum_{n\ge1}n^{-s} \qquad(\operatorname{Re}s>1)$$ is the Riemann zeta function.
For a real transcendental number $\xi$, let $\omega_n^*(\xi)$ denote the supremum of all $\omega$ for which there exist infinitely many real algebraic numbers $\alpha$ of degree $\leq n$ satisfying $|\xi-\alpha|\leq H(\alpha)^{-\omega -1}$, where $H(\alpha)$ is the naive height of the minimal polynomial of $\alpha$. A celebrated result of Wirsing gives the uniform lower bound $\omega_n^*(\xi)\geq\frac{n+1}{2}$, which was improved significantly in a recent work of Po\"els to $\frac{n}{2-\log 2}$. In this paper, we establish a $p$-adic counterpart of Po\"els's result. Let $p$ be a prime and $\xi\in\Qp$ be transcendental. Let $\omega_{n,p}^*(\xi)$ be the supremum of all real numbers $\omega$ for which there exist infinitely many algebraic numbers $\alpha \in \Qp$ of degree $\leq n$ such that $|\xi-\alpha|_p\leq H(\alpha)^{-\omega -1}$. We show that $\omega^*_{n,p}(\xi)\geq\frac{n}{2-\log 2}-1$. This improves the known lower bounds in the $p$-adic setting, namely the analogue of Wirsing's theorem, due to Morrison and Teuli\'e.
For every finite family of Euclidean balls in $\mathbb{R}^n$, $n\ge2$, and every $0<\delta<1/2$, we select a subfamily whose $(1+\delta)$-dilations cover the original union and whose undilated balls have multiplicity at most $C_n\delta^{-(n-1)/2}$. This proves the covering estimate conjectured by Han and Lu \cite{HL}. As an application, we determine the optimal Solyanik asymptotic \[ \mathcal C_n(\alpha)-1\asymp_n(1-\alpha)^{2/(n+1)} \qquad(\alpha\uparrow1) \] for the uncentered Hardy--Littlewood maximal operator over Euclidean balls, which is Conjecture~1(b) of Hagelstein and Parissis \cite{HP14}. For the modified uncentered maximal operators, we determine the optimal weak $(1,1)$ growth rate $(k-1)^{-(n-1)/2}$ as $k\downarrow1$, uniformly over Radon measures. We also obtain the corresponding $L^p$ and Fefferman--Stein bounds.
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