Niclas Technau
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View article: Full Poissonian local statistics of slowly growing sequences
Full Poissonian local statistics of slowly growing sequences Open
Fix $\alpha >0$ . Then by Fejér's theorem $(\alpha (\log n)^{A}\,\mathrm {mod}\,1)_{n\geq 1}$ is uniformly distributed if and only if $A>1$ . We sharpen this by showing that all correlation functions, and hence the gap distribution, are Po…
View article: Smooth discrepancy and Littlewood's conjecture
Smooth discrepancy and Littlewood's conjecture Open
Given $\boldsymbolα \in [0,1]^d$, we estimate the smooth discrepancy of the Kronecker sequence $(n \boldsymbolα \,\mathrm{mod}\, 1)_{n\geq 1}$. We find that it can be smaller than the classical discrepancy of $\textbf{any}$ sequence when $…
View article: Dispersion and Littlewood's conjecture
Dispersion and Littlewood's conjecture Open
Let ε>0. We construct an explicit, full-measure set of α∈[0,1] such that if γ∈R then, for almost all β∈[0,1], if δ∈R then there are infinitely many integers n⩾1 for whichn‖nα−γ‖⋅‖nβ−δ‖<(loglogn)3+εlogn. This is a significant quantitativ…
View article: Pair correlation of the fractional parts of $\alpha n^{\theta}$
Pair correlation of the fractional parts of $\alpha n^{\theta}$ Open
Fix \alpha,\theta >0 and consider the sequence (\alpha n^{\theta}\; \mathrm{mod}\; 1)_{n\ge 1} . Since the seminal work of Rudnick–Sarnak (1998) and due to the Berry–Tabor conjecture in quantum chaos, the fine-scale properties of these dil…
View article: Gap distribution of $\sqrt{n} \,\mathrm{mod}\, 1$ and the circle method
Gap distribution of $\sqrt{n} \,\mathrm{mod}\, 1$ and the circle method Open
The distribution of the properly renormalized gaps of $\sqrt{n} \,\mathrm{mod}\, 1$ with $n < N$ converges (when $N\rightarrow \infty$) to a non-standard limit distribution, as Elkies and McMullen proved in 2004 using techniques from homog…
View article: Rational Points Near Manifolds, Homogeneous Dynamics, and Oscillatory Integrals
Rational Points Near Manifolds, Homogeneous Dynamics, and Oscillatory Integrals Open
Let $\mathcal{M}\subset \mathbb{R}^n$ be a compact and sufficiently smooth manifold of dimension $d$. Suppose $\mathcal{M}$ is nowhere completely flat. Let $N_{\mathcal{M}}(δ,Q)$ denote the number of rational vectors $\mathbf{a}/q$ within …
View article: Dispersion and Littlewood's conjecture
Dispersion and Littlewood's conjecture Open
Let $\varepsilon>0$. We construct an explicit, full-measure set of $α\in[0,1]$ such that if $γ\in \mathbb{R}$ then, for almost all $β\in[0,1]$, if $δ\in \mathbb{R}$ then there are infinitely many integers $n\geq 1$ for which \[ n \Vert nα-…
View article: On the order of magnitude of Sudler products
On the order of magnitude of Sudler products Open
Given an irrational number $\alpha\in(0,1)$, the Sudler product is defined by $P_N(\alpha) = \prod_{r=1}^{N}2|\sin\pi r\alpha|$. Answering a question of Grepstad, Kaltenb\"ock and Neum\"uller we prove an asymptotic formula for distorted Su…
View article: Density of Rational Points Near Flat/Rough Hypersurfaces
Density of Rational Points Near Flat/Rough Hypersurfaces Open
For $n\geq 3$, let $\mathscr{M} \subseteq\mathbb{R}^{n}$ be a compact hypersurface, parametrized by a homogeneous function of degree $d\in \mathbb{R}_{>1}$, with non-vanishing curvature away from the origin. Consider the number $\mathrm{N}…
View article: On the correlations of $n^\alpha$ mod 1
On the correlations of $n^\alpha$ mod 1 Open
A well known result in the theory of uniform distribution modulo 1 (which goes back to Fejér and Csillag) states that the fractional parts \{n^{\alpha}\} of the sequence (n^{\alpha})_{n\ge1} are uniformly distributed in the unit interval w…
View article: Full Poissonian Local Statistics of Slowly Growing Sequences
Full Poissonian Local Statistics of Slowly Growing Sequences Open
Fix $α>0$, then by Fejér's theorem $ (α(\log n)^{A}\,\mathrm{mod}\,1)_{n\geq1}$ is uniformly distributed if and only if $A>1$. We sharpen this by showing that all correlation functions, and hence the gap distribution, are Poissonian provid…
View article: Counting multiplicative approximations
Counting multiplicative approximations Open
A famous conjecture of Littlewood (c. 1930) concerns approximating two real numbers by rationals of the same denominator, multiplying the errors. In a lesser-known paper, Wang and Yu (1981) established an asymptotic formula for the number …
View article: Correlations of the Fractional Parts of $αn^θ$
Correlations of the Fractional Parts of $αn^θ$ Open
Let $m\geq 3$, we prove that $(αn^θ\mod 1)_{n>0}$ has Poissonian $m$-point correlation for all $α>0$, provided $θ
View article: Northcott numbers for the house and the Weil height
Northcott numbers for the house and the Weil height Open
For an algebraic number $α$ and $γ\in \mathbb{R}$, $h(α)$ be the (logarithmic) Weil height, and $h_γ(α)=(\mathrm{deg}α)^γh(α)$ be the $γ$-weighted (logarithmic) Weil height of $α$. Let $f:\overline{\mathbb{Q}}\to [0,\infty)$ be a function …
View article: The metric theory of the pair correlation function for small non-integer powers
The metric theory of the pair correlation function for small non-integer powers Open
For $0
View article: Pair Correlation of the Fractional Parts of $αn^θ$
Pair Correlation of the Fractional Parts of $αn^θ$ Open
Fix $α,θ>0$, and consider the sequence $(αn^θ \mod 1)_{n\ge 1}$. Since the seminal work of Rudnick--Sarnak (1998), and due to the Berry--Tabor conjecture in quantum chaos, the fine-scale properties of these dilated mononomial sequences hav…
View article: Lehmer without Bogomolov
Lehmer without Bogomolov Open
We construct fields of algebraic numbers that have the Lehmer property but not the Bogomolov property. This answers a recent implicit question of Pengo and the first author.
View article: Gap statistics and higher correlations for geometric progressions modulo\n one
Gap statistics and higher correlations for geometric progressions modulo\n one Open
Koksma's equidistribution theorem from 1935 states that for Lebesgue almost\nevery $\\alpha>1$, the fractional parts of the geometric progression\n$(\\alpha^{n})_{n\\geq1}$ are equidistributed modulo one. In the present paper we\nsharpen t…
View article: Littlewood and Duffin--Schaeffer-type problems in diophantine approximation
Littlewood and Duffin--Schaeffer-type problems in diophantine approximation Open
Gallagher's theorem describes the multiplicative diophantine approximation rate of a typical vector. We establish a fully-inhomogeneous version of Gallagher's theorem, a diophantine fibre refinement, and a sharp and unexpected threshold fo…
View article: On the correlations of $n^α$ mod 1
On the correlations of $n^α$ mod 1 Open
A well known result in the theory of uniform distribution modulo one (which goes back to Fejér and Csillag) states that the fractional parts $\{n^α\}$ of the sequence $(n^α)_{n\ge1}$ are uniformly distributed in the unit interval whenever …
View article: On the triple correlations of fractional parts of $n^2α$
On the triple correlations of fractional parts of $n^2α$ Open
For fixed $α\in [0,1]$, consider the set $S_{α,N}$ of dilated squares $α, 4α, 9α, \dots, N^2α\, $ modulo $1$. Rudnick and Sarnak conjectured that for Lebesgue almost all such $α$ the gap-distribution of $S_{α,N}$ is consistent with the Poi…
View article: On the triple correlations of fractional parts of $n^2\alpha$
On the triple correlations of fractional parts of $n^2\alpha$ Open
For fixed $\\alpha \\in [0,1]$, consider the set $S_{\\alpha,N}$ of dilated\nsquares $\\alpha, 4\\alpha, 9\\alpha, \\dots, N^2\\alpha \\, $ modulo $1$. Rudnick\nand Sarnak conjectured that for Lebesgue almost all such $\\alpha$ the\ngap-di…
View article: On the order of magnitude of Sudler products
On the order of magnitude of Sudler products Open
Given an irrational number $α\in(0,1)$, the Sudler product is defined by $P_N(α) = \prod_{r=1}^{N}2|\sinπrα|$. Answering a question of Grepstad, Kaltenböck and Neumüller we prove an asymptotic formula for distorted Sudler products when $α$…
View article: ON LINEAR RELATIONS FOR DIRICHLET SERIES FORMED BY RECURSIVE SEQUENCES OF SECOND ORDER
ON LINEAR RELATIONS FOR DIRICHLET SERIES FORMED BY RECURSIVE SEQUENCES OF SECOND ORDER Open
Let $F_{n}$ and $L_{n}$ be the Fibonacci and Lucas numbers, respectively. Four corresponding zeta functions in $s$ are defined by $$\begin{eqnarray}\unicode[STIX]{x1D701}_{F}(s):=\mathop{\sum }_{n=1}^{\infty }{\displaystyle \frac{1}{F_{n}^…
View article: THERE IS NO KHINTCHINE THRESHOLD FOR METRIC PAIR CORRELATIONS
THERE IS NO KHINTCHINE THRESHOLD FOR METRIC PAIR CORRELATIONS Open
We consider sequences of the form $\\left(a_{n} \\alpha\\right)_{n}$ mod 1,\nwhere $\\alpha\\in\\left[0,1\\right]$ and where $\\left(a_{n}\\right)_{n}$ is a\nstrictly increasing sequence of positive integers. If the asymptotic\ndistributio…
View article: The discrepancy of $(n_kx)$ with respect to certain probability measures
The discrepancy of $(n_kx)$ with respect to certain probability measures Open
Let $(n_k)_{k=1}^{\infty}$ be a lacunary sequence of integers. We show that if $μ$ is a probability measure on $[0,1)$ such that $|\widehatμ(t)|\leq c|t|^{-η}$, then for $μ$-almost all $x$, the discrepancy $D_N(n_kx)$ satisfies \begin{equa…
View article: On the regularity of primes in arithmetic progressions
On the regularity of primes in arithmetic progressions Open
We prove that for a positive integer [Formula: see text] the primes in certain kinds of intervals cannot distribute too “uniformly” among the reduced residue classes modulo [Formula: see text]. Hereby, we prove a generalization of a conjec…
View article: On a Counting Theorem of Skriganov
On a Counting Theorem of Skriganov Open
We prove a counting theorem concerning the number of lattice points for the dual lattices of weakly admissible lattices in an inhomogeneously expanding box, which generalises a counting theorem of Skriganov. The error term is expressed in …