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.
In this note, we give a counterexample to the assertion of Problem 10.4 in Bugeaud's monograph Distribution modulo one and Diophantine approximation, which goes back to Mend\`es France. The problem states that the spectrum of the sequence $(\xi\alpha^n)_{n\ge1}$, that is, the set of irrational $\theta\in(0,1)$ for which $(\xi\alpha^n-n\theta)_{n\ge1}$ is not uniformly distributed modulo one, is at most countable for all real $\xi\ne0$ and $\alpha>1$. More precisely, we prove that for every real $\alpha>1$ there are $2^{\aleph_0}$ real numbers $\xi>0$ for which the spectrum of $(\xi\alpha^n)_{n\ge1}$ contains one and the same uncountable set.
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 the minor arcs of the Hardy--Littlewood dissection throughout the range $Q\le N^{1/2}$. The estimates of Schlage-Puchta [SP] and of Tolev [T] have the same dependence on $q$ and $Q$ but carry a factor $N^{\varepsilon}$. The proof uses Heath-Brown's square sieve with sieving primes confined to an interval $(P,2P]$, where $P$ may be as small as a multiple of $\log N$; a finite Fej\'er majorant in place of a truncated Fourier series; and, after completion of the character sums, a count of representations that exploits the restriction on the primes in place of the divisor function.
Nicolas Robles, Alexandru Zaharescu, Dirk Zeindler· 0 citations
Let $\pi_n$ be the monic polynomial of degree $n$ orthogonal on $[-c,c]$, $0<c\leq 1$, with respect to the Jacobi weight $(1-x)^\alpha(1+x)^\beta$, where $-1<\alpha<\beta$. Gautschi conjectured that $$ \left[ \frac{\pi_n(-c)}{\pi_n(c)} \right]^2 \left(\frac{1-c}{1+c}\right)^{\beta-\alpha}<1. $$ By his variation formula, this inequality is sufficient for every positive zero of $\pi_n$ to move to the right as $c$ increases. For $0<c<1$, we prove the conjecture, uniformly in the degree, throughout $0<\alpha<\beta$. Combined with the region $\alpha\leq 0\leq\beta$, recorded by Gautschi on the basis of an unpublished communication from Milovanovi'c, and with Milovanovi'c's published criterion, this settles the full admissible range $\beta\geq 0$. In the remaining negative wedge, writing $\alpha=-r-\lambda$ and $\beta=-r+\lambda$, we prove the conjecture whenever $$ c^2\leq\frac{3}{3+r}. $$ Consequently, it holds for every admissible pair of parameters when $0<c\leq\sqrt{3}/2$. For arbitrary admissible parameters, we also establish the degree-one case and eventual validity as $n\to\infty$. The proof combines an exact boundary identity, a first-crossing argument based on Pearson and root-motion identities, and, in the negative wedge, a Ward identity with positive association for an MTP$_2$ orthogonal polynomial ensemble. The case $c=1$ is immediate.
V. Botta, K. Castillo, L. Tertuliano et al.· 0 citations
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 $\alpha \in \mathbb{R} \setminus \mathbb{Q}$, $\beta \in \R$, $N \in \mathbb{R}_{\ge 1}$ and $ \Delta \in (0, 1/2)$. For any real $y$, let $\|y\|$ denote the distance from $y$ to the nearest integer. In the first part of this paper, we show that given two coprime integers $u, v \ge 1$, there are infinitely many primes $\ell \equiv u \bmod v$ such that $ \|\alpha \ell - \beta\| \ll_v \ell^{-1/4} \log^{8} \ell. $ In order to prove this result we first prove the following general theorem and then deduce the above as a corollary. Before stating the result, let us define a function on $f_{\Delta}(\theta)$ on $\R$ such that $f_{\Delta}(\theta)$ is $ 1 \text{ if } \| \theta \|<\Delta$ and $ 0 $ otherwise. Further, suppose that $u, v \in \mathbb{Z}_{\ge 1}$ are coprime and $a$, $q \in \Z$ are coprime with $q>N^{1/4}$ and $|\alpha v -a/q| \le 1/q^2$. Then, for every $\epsilon \in \mathbb{R}_{>0}$ we have \begin{equation*} \sum_{\substack{n=1 \\ n \equiv u \bmod v}}^N \Lambda(n) (f_{\Delta}(\alpha n - \beta) - 2\Delta) \ll_v (Nq^{-1/2} + N^{3/4} + N^{5/6}\Delta^{1/2} + (\Delta Nq)^{1/2} + N^{\epsilon}q \Delta^{1-\epsilon}) \mathcal{L}^8 \end{equation*} where $\mathcal{L}=\log (Nq/\Delta)$. This generalises a well known result of Vaughan from 1977 proving a similar bound for the sum \begin{equation*} \sum_{\substack{n=1}}^N \Lambda(n) (f_{\Delta}(\alpha n - \beta) - 2\Delta). \end{equation*} In the second part of this paper, we explicitly construct an uncountable set $S \subset \R\setminus \Q$ such that for every $\gamma \in S$ there are infinitely many primes $\ell \equiv u \bmod v$ satisfying $\| \gamma \ell \|<\ell^{-1}$. Further we prove unconditionally that not all the elements of $S$ are Liouville numbers. This addresses a question of Erd{\"o}s and Mahler from 1939.
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.
Adam Kanigowski, Kaitlyn Loyd· 0 citations
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