Niclas Technau

activity

20162026

most citedThere is no Khintchine threshold for metric pair correlations

15 citations · 53 across the 27 of their papers we have counts for

collaborators

31 papers

math.NT2026

Counterexamples for Rational Points Near Curves

Mingfeng Chen, Rajula Srivastava, Niclas Technau

Let $δ \in [0, 1/2]$ and $Q \geq 1$ . Given a compact $C^{\infty}$ -curve $C$ in $R^{n}$ , denote by $N_{C} (δ, Q)$ the number of pairs $(\mathbf{a}, q)\in \mat…

math.NT2026

Sharp Bounds for Rational Points Near Space Curves

Mingfeng Chen, Andreas Seeger, Rajula Srivastava +1

Let $Q \geq 1$ be large, and $δ \in (0, 1)$ be small. Denote by $C \subset R^{3}$ a sufficiently smooth curve with non-vanishing curvature and torsion. How many rational…

math.NT2026

Poissonian correlations of $α n^{d}$ mod $1$

Chris Lutsko, Nick Rome, Niclas Technau

Let $x (n) := α n^{d} mod 1$ for integer $d > 1$ and non-zero real $α$ . We show that ${x (n)}_{n > 0}$ has Poissonian $ℓ$ -point correlations for almost all choices of $α$ when $d$ is l…

math.NT2026

Rational Points in Hyperbolic Regions and Multiplicative Diophantine Approximation on Manifolds

Sam Chow, Rajula Srivastava, Niclas Technau +1

We establish the convergence theory of multiplicative Diophantine approximation for all non-degenerate, smooth manifolds. We also settle said convergence theory for all affine subs…

math.NT2024

Smooth discrepancy and Littlewood's conjecture

Sam Chow, Niclas Technau

Given $α \in [0, 1]^{d}$ , we estimate the smooth discrepancy of the Kronecker sequence $(n α mod 1)_{n \geq 1}$ . We find that it can be smaller than…

math.NT2024

Gap distribution of $n mod 1$ and the circle method

Maksym Radziwiłł, Niclas Technau

The distribution of the properly renormalized gaps of $n mod 1$ with $n < N$ converges (when $N \to \infty$ ) to a non-standard limit distribution, as Elk…

math.NT2023★ 1 cited

Rational Points Near Manifolds, Homogeneous Dynamics, and Oscillatory Integrals

Damaris Schindler, Rajula Srivastava, Niclas Technau

Let $M \subset R^{n}$ be a compact and sufficiently smooth manifold of dimension $d$ . Suppose $M$ is nowhere completely flat. Let $N_{M} (δ, Q)$ d…

math.NT2023

Dispersion and Littlewood's conjecture

Sam Chow, Niclas Technau

Let $ε > 0$ . We construct an explicit, full-measure set of $α \in [0, 1]$ such that if $γ \in R$ then, for almost all $β \in [0, 1]$ , if $δ \in R$ then there are…

math.NT2023★ 2 cited

Density of Rational Points Near Flat/Rough Hypersurfaces

Rajula Srivastava, Niclas Technau

For $n \geq 3$ , let $M \subseteq R^{n}$ be a compact hypersurface, parametrized by a homogeneous function of degree $d \in R_{> 1}$ , with non-vanishing curv…

math.NT2022★ 1 cited

Full Poissonian Local Statistics of Slowly Growing Sequences

Christopher Lutsko, Niclas Technau

Fix $α > 0$ , then by Fejér's theorem $(α (lo g n)^{A} mod 1)_{n \geq 1}$ is uniformly distributed if and only if $A > 1$ . We sharpen this by showing that all correlation funct…

math.NT2022

Counting multiplicative approximations

Sam Chow, Niclas Technau

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…

math.NT2021

Correlations of the Fractional Parts of $α n^{θ}$

Christopher Lutsko, Niclas Technau

Let $m \geq 3$ , we prove that $(α n^{θ} mod 1)_{n > 0}$ 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 $m$ in…

math.NT2021

The metric theory of the pair correlation function for small non-integer powers

Zeév Rudnick, Niclas Technau

For $0 < θ < 1$ , we show that for almost all $α$ , the pair correlation function of the sequence of fractional parts of ${α n^{θ} : n \geq 1}$ is Poissonian.

math.NT2021

Northcott numbers for the house and the Weil height

Fabien Pazuki, Niclas Technau, Martin Widmer

For an algebraic number $α$ and $γ \in R$ , $h (α)$ be the (logarithmic) Weil height, and $h_{γ} (α) = (deg α)^{γ} h (α)$ be the $γ$ -weighted (logarithmic) Weil height of $α$ .…

math.NT2021★ 1 cited

Pair Correlation of the Fractional Parts of $α n^{θ}$

Christopher Lutsko, Athanasios Sourmelidis, Niclas Technau

Fix $α, θ > 0$ , and consider the sequence $(α n^{θ} mod 1)_{n \geq 1}$ . Since the seminal work of Rudnick--Sarnak (1998), and due to the Berry--Tabor conjecture in quantum chaos, the fine…

math.NT2020

Lehmer without Bogomolov

Fabien Pazuki, Niclas Technau, Martin Widmer

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.

math.NT2020★ 6 cited

Gap statistics and higher correlations for geometric progressions modulo one

Christoph Aistleitner, Simon Baker, Niclas Technau +1

Koksma's equidistribution theorem from 1935 states that for Lebesgue almost every $α > 1$ , the fractional parts of the geometric progression $(α^{n})_{n \geq 1}$ are equidistributed mo…

math.NT2020★ 5 cited

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

Sam Chow, Niclas Technau

Gallagher's theorem describes the multiplicative diophantine approximation rate of a typical vector. We establish a fully-inhomogeneous version of Gallagher's theorem, a diophantin…

math.NT2020

On the correlations of $n^{α}$ mod 1

Niclas Technau, Nadav Yesha

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…

math.NT2020

On the triple correlations of fractional parts of $n^{2} α$

Niclas Technau, Aled Walker

For fixed $α \in [0, 1]$ , consider the set $S_{α, N}$ of dilated squares $α, 4 α, 9 α, \dots, N^{2} α$ modulo $1$ . Rudnick and Sarnak conjectured that for Lebesgue almost all such $α$ t…

math.NT2020★ 1 cited

On the order of magnitude of Sudler products

Christoph Aistleitner, Niclas Technau, Agamemnon Zafeiropoulos

Given an irrational number $α \in (0, 1)$ , the Sudler product is defined by $P_{N} (α) = \prod_{r = 1}^{N} 2∣ sin π r α ∣$ . Answering a question of Grepstad, Kaltenböck and Neumüller we prove a…

math.NT2020★ 6 cited

The metric theory of the pair correlation function of real-valued lacunary sequences

Niclas Technau, Zeév Rudnick

Let ${a (x)}_{x = 1}^{\infty}$ be a positive, real-valued, lacunary sequence. This note shows that the pair correlation function of the fractional parts of the dilations $α a (x)$ i…

math.NT2018

The discrepancy of $(n_{k} x)$ with respect to certain probability measures

Niclas Technau, Agamemnon Zafeiropoulos

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 $∣ μ (t) ∣ \leq c ∣ t ∣^{- η}$ , then for $μ$ -almost…

math.NT2018★ 11 cited

Higher-rank Bohr sets and multiplicative diophantine approximation

Sam Chow, Niclas Technau

Gallagher's theorem is a sharpening and extension of the Littlewood conjecture that holds for almost all tuples of real numbers. We provide a fibre refinement, solving a problem po…

Counterexamples for Rational Points Near Curves

Sharp Bounds for Rational Points Near Space Curves

Poissonian correlations of αnd mod 1

Rational Points in Hyperbolic Regions and Multiplicative Diophantine Approximation on Manifolds

Smooth discrepancy and Littlewood's conjecture

Gap distribution of n​mod1 and the circle method

Rational Points Near Manifolds, Homogeneous Dynamics, and Oscillatory Integrals

Dispersion and Littlewood's conjecture

Density of Rational Points Near Flat/Rough Hypersurfaces

Full Poissonian Local Statistics of Slowly Growing Sequences

Counting multiplicative approximations

Correlations of the Fractional Parts of αnθ

The metric theory of the pair correlation function for small non-integer powers

Northcott numbers for the house and the Weil height

Pair Correlation of the Fractional Parts of αnθ

Lehmer without Bogomolov

Gap statistics and higher correlations for geometric progressions modulo one

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

On the correlations of nα mod 1

On the triple correlations of fractional parts of n2α

On the order of magnitude of Sudler products

The metric theory of the pair correlation function of real-valued lacunary sequences

The discrepancy of (nk​x) with respect to certain probability measures

Higher-rank Bohr sets and multiplicative diophantine approximation

Poissonian correlations of $α n^{d}$ mod $1$

Gap distribution of $n mod 1$ and the circle method

Correlations of the Fractional Parts of $α n^{θ}$

Pair Correlation of the Fractional Parts of $α n^{θ}$

On the correlations of $n^{α}$ mod 1

On the triple correlations of fractional parts of $n^{2} α$

The discrepancy of $(n_{k} x)$ with respect to certain probability measures