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Niclas Technau

Publications and source records attributed to Niclas Technau.

At least 19 recordsLinked to original sources
arXiv2026

Height lower bounds for elements of highly composite rings

Let $\mathbb{Q}^{(d)}$ be the composite field of all number fields of degree at most $d$. In 2001 Bombieri and Zannier proved that $\mathbb{Q}^{(2)}$ has the Northcott property and asked what happens for $d\geq 3$. Here we study the absolute Weil height for elements in the composite ring of the rings of integers of such number fields. In particular, we consider $\mathbb{Q}^{(3)}$ as the composite field of $\mathbb{Q}^{(2)}$ and a minimal infinite family of cubic fields, and we show that the composite ring of the rings of integers of these fields does have the Northcott property. Our results follow from new height lower bounds, expressed in terms of the degree. Moreover, we introduce a notion of size for subfields of $\mathbb{Q}^{(3)}$. For instance, $\mathbb{Q}^{(3)}$ has size $1$ and the maximal abelian subfield of $\mathbb{Q}^{(3)}$ has size $1/2$. We show that there is a subfield of $\mathbb{Q}^{(3)}$ of size $1$ which has the Northcott property.

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arXiv2026

Sharp Bounds for Rational Points Near Space Curves

Let $Q\geq 1$ be large, and $\delta \in(0,1)$ be small. Denote by $\mathcal C \subset \mathbb R^3$ a sufficiently smooth curve with non-vanishing curvature and torsion. How many rational points $\mathbf{a}/q$ of height $q\in[1, Q]$ are $\delta/q$-near $\mathcal C$? This manuscript provides an essentially optimal answer, thereby addressing a problem stated by Beresnevich and Kleinbock, for space curves. We show that the folklore conjectures are incorrect for certain manifolds with codimension $\ge 2$, including the moment curve $(t,t^2,t^3)$. The reason is a {hitherto} hidden `major arc' type obstruction. We also establish matching upper bounds, up to endpoints. Our argument combines purely Fourier analytical techniques with the planar counting results by Vaughan and Velani.

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arXiv2026

Poissonian correlations of $\alpha n^d$ mod $1$

Let $x(n):=\alpha n^d \mod 1$ for integer $d >1$ and non-zero real $\alpha$. We show that $\{x(n)\}_{n>0}$ has Poissonian $\ell$-point correlations for almost all choices of $\alpha$ when $d$ is large (depending on $\ell$). This falls in line with the expected behavior from the Berry--Tabor conjecture. Further, in the spirit of a conjecture of Rudnick--Sarnak, we show Poissonian $\ell$-point correlations for a set of badly approximable $\alpha$ of full Hausdorff dimension by a Fourier analytic transference principle. The proof makes use of an application of the determinant method to count points on a diagonal hypersurface of degree $d$ in such a way as to capture the contribution of points belonging to lower dimensional varieties. As $d$ grows, these `special solutions' dominate the count and non-special solutions become increasingly rare. This stratified counting statement allows us to control the number of points on average very effectively.

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arXiv2026

Rational Points in Hyperbolic Regions and Multiplicative Diophantine Approximation on Manifolds

We establish the convergence theory of multiplicative Diophantine approximation for all non-degenerate, smooth manifolds. We also settle said convergence theory for all affine subspaces satisfying a highly generic and essentially optimal Diophantine condition. This answers a question of Beresnevich and Velani from 2005, while simultaneously sharpening results of Kleinbock and Margulis on the strong extremality of non-degenerate manifolds, and of Kleinbock on the strong extremality of affine subspaces.

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arXiv2024

Smooth discrepancy and Littlewood's conjecture

Given $\boldsymbol{\alpha} \in [0,1]^d$, we estimate the smooth discrepancy of the Kronecker sequence $(n \boldsymbol{\alpha} \,\mathrm{mod}\, 1)_{n\geq 1}$. We find that it can be smaller than the classical discrepancy of $\textbf{any}$ sequence when $d \le 2$, and can even be bounded in the case $d=1$. To achieve this, we establish a novel deterministic analogue of Beck's local-to-global principle (Ann. of Math. 1994), which relates the discrepancy of a Kronecker sequence to multiplicative diophantine approximation. This opens up a new avenue of attack for Littlewood's conjecture.

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arXiv2024

Gap distribution of $\sqrt{n} \,\mathrm{mod}\, 1$ and the circle method

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 homogeneous dynamics. In this paper we give an essentially self-contained proof based on the circle method. Our main innovation consists in showing that a new type of correlation functions of $\sqrt{n} \,\mathrm{mod}\, 1$ converge. To define these correlation functions we restrict, smoothly, to those $\sqrt{n} \,\mathrm{mod}\, 1$ that lie in minor arcs, i.e. away from rational numbers with small denominators.

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arXiv2023

Rational Points Near Manifolds, Homogeneous Dynamics, and Oscillatory Integrals

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 $N_{\mathcal{M}}(\delta,Q)$ denote the number of rational vectors $\mathbf{a}/q$ within a distance of $\delta/q$ from $\mathcal{M}$ so that $q \in [Q,2Q)$. We develop a novel method to analyse $N_{\mathcal{M}}(\delta,Q)$. The salient feature of our technique is the combination of powerful quantitative non-divergence estimates, in a form due to Bernik, Kleinbock, and Margulis, with Fourier analytic tools. The second ingredient enables us to eschew the Dani correspondence and an explicit use of the geometry of numbers. We employ this new method to address in a strong sense a problem of Beresnevich regarding lower bounds on $N_{\mathcal{M}}(\delta,Q)$ for non-analytic manifolds. Additionally, we obtain asymptotic formulae which are the first of their kind for such a general class of manifolds. As a by-product, we improve upon upper bounds on $N_{\mathcal{M}}(\delta,Q)$ from a recent breakthrough of Beresnevich and Yang and recover their convergence Khintchine type theorem for arbitrary nondegenerate submanifolds. Moreover, we obtain new Hausdorff dimension and measure refinements for the set of well-approximable points for a range of Diophantine exponents close to $1/n$.

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arXiv2023

Littlewood and Duffin--Schaeffer-type problems in diophantine approximation

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 for Liouville fibres. Along the way, we prove an inhomogeneous version of the Duffin--Schaeffer conjecture for a class of non-monotonic approximation functions.

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arXiv2023

Dispersion and Littlewood's conjecture

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 \[ n \Vert nα- γ\Vert \cdot \Vert nβ- δ\Vert < \frac{(\log \log n)^{3 + \varepsilon}}{\log n}. \] This is a significant quantitative improvement over a result of the first author and Zafeiropoulos. We show, moreover, that the exceptional set of $β$ has Fourier dimension zero, alongside further applications to badly approximable numbers and to lacunary diophantine approximation. Our method relies on a dispersion estimate and the Three Distance Theorem.

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arXiv2023

Density of Rational Points Near Flat/Rough Hypersurfaces

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 $\mathrm{N}_{\mathscr{M}}(\delta,Q)$ of rationals $\mathbf{a}/q$, with denominator $q\in [Q,2Q)$ and $\mathbf{a} \in \mathbb{Z}^{n-1}$, lying at a distance at most $\delta/q$ from $\mathscr{M}$. This manuscript provides essentially sharp estimates for $\mathrm{N}_{\mathscr{M}}(\delta,Q)$ throughout the range $\delta \in (Q^{\varepsilon-1},1/2)$ for $d>1+\tfrac{1}{2n-3}$. Our result is a first of its kind for hypersurfaces with vanishing Gaussian curvature ($d>2$) and those which are rough (meaning not even $C^2$ at the origin which happens when $d<2$). An interesting outcome of our investigation is the understanding of a `geometric' term $(\delta/Q)^{(n-1)/d}Q^n$ (stemming from a so-called Knapp cap), arising in addition to the usual probabilistic term $\delta Q^n$; the sum of these terms determines the size of $\mathrm{N}_{\mathscr{M}}(\delta,Q)$ for $\delta\in(Q^{\varepsilon-1},1/2)$. Consequences of our result concern the metric theory of Diophantine approximation on `rough' hypersurfaces -- going beyond the recent break-through of Beresnevich and L. Yang. Further, we establish smooth extensions of Serre's dimension growth conjecture.

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arXiv2023

Pair Correlation of the Fractional Parts of $αn^θ$

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 have been intensively studied. In this paper we show that for $θ\le 1/3$, and $α>0$, the pair correlation function is Poissonian. While (for a given $θ\neq 1$) this strong pseudo-randomness property has been proven for almost all values of $α$, there are next-to-no instances where this has been proven for explicit $α$. Our result holds for all $α>0$ and relies solely on classical Fourier analytic techniques. This addresses (in the sharpest possible way) a problem posed by Aistleitner--El-Baz--Munsch (2021).

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arXiv2022

Full Poissonian Local Statistics of Slowly Growing Sequences

Fix $\alpha>0$, then by Fej\'er's theorem $ (\alpha(\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 provided $A>1$. This is the first example of a deterministic sequence modulo one whose gap distribution, and all of whose correlations are proven to be Poissonian. The range of $A$ is optimal and complements a result of Marklof and Str\"{o}mbergsson who found the limiting gap distribution of $(\log(n)\, \mathrm{mod}\,1)$, which is necessarily not Poissonian.

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arXiv2022

Counting multiplicative approximations

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 of such approximations, valid almost always. Using the quantitative Koukoulopoulos--Maynard theorem of Aistleitner--Borda--Hauke, together with bounds arising from the theory of Bohr sets, we deduce lower bounds of the expected order of magnitude for inhomogeneous and fibre refinements of the problem.

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arXiv2022

Northcott numbers for the house and the Weil height

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 on the algebraic numbers $\overline{\mathbb{Q}}$, and let $S\subset \overline{\mathbb{Q}}$. The Northcott number $\mathcal{N}_f(S)$ of $S$, with respect to $f$, is the infimum of all $X\geq 0$ such that $\{α\in S; f(α)< X\}$ is infinite. This paper studies the set of Northcott numbers $\mathcal{N}_f(\mathcal{O})$ for subrings of $\overline{\mathbb{Q}}$ for the house, the Weil height, and the $γ$-weighted Weil height. We show: (1) Every $t\geq 1$ is the Northcott number of a ring of integers of a field w.r.t. the house. (2) For each $t\geq 0$ there exists a field with Northcott number in $ [t,2t]$ w.r.t. the Weil height $h(\cdot)$. (3) For all $0\leq γ\leq 1$ and $γ'<γ$ there exists a field $K$ with $\mathcal{N}_{h_{γ'}}(K)=0$ and $\mathcal{N}_{h_γ}(K)=\infty$. For $(1)$ we provide examples that satisfy an analogue of Julia Robinon's property (JR), examples that satisfy an analogue of Vidaux and Videla's isolation property, and examples that satisfy neither of those. Item $(2)$ concerns a question raised by Vidaux and Videla due to its direct link with decidability theory via the Julia Robinson number. Item (3) is a strong generalisation of the known fact that there are fields that satisfy the Lehmer conjecture but which are not Bogomolov in the sense of Bombieri and Zannier.

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arXiv2022

On the order of magnitude of Sudler products

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 $α$ is the golden ratio $(\sqrt{5}+1)/2$ and establish that in this case $\limsup_{N \to \infty} P_N(α)/N < \infty$. We obtain similar results for quadratic irrationals $α$ with continued fraction expansion $α= [a,a,a,\dots]$ for some integer $a \geq 1$, and give a full characterization of the values of $a$ for which $\liminf_{N \to \infty} P_N(α)>0$ and $\limsup_{N \to \infty} P_N(α) / N < \infty$ hold, respectively. We establish that there is a (sharp) transition point at $a=6$, and resolve as a by-product a problem of the first named author, Larcher, Pillichshammer, Saad Eddin, and Tichy.

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arXiv2021

Correlations of the Fractional Parts of $αn^θ$

Let $m\geq 3$, we prove that $(αn^θ\mod 1)_{n>0}$ has Poissonian $m$-point correlation for all $α>0$, provided $θ<θ_m$, where $θ_m$ is an explicit bound which goes to $0$ as $m$ increases. This work builds on the method developed in Lutsko-Sourmelidis-Technau (2021), and introduces a new combinatorial argument for higher correlation levels, and new Fourier analytic techniques. A key point is to introduce an `extra' frequency variable to de-correlate the sequence variables and to eventually exploit a repulsion principle for oscillatory integrals. Presently, this is the only positive result showing that the $m$-point correlation is Poissonian for such sequences.

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arXiv2021

Lehmer without Bogomolov

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.

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