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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arXiv subjects
Niclas Technau
Publications and source records attributed to Niclas Technau.
On the triple correlations of fractional parts of $n^2α$
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 Poisson model (in the limit as $N$ tends to infinity). In this paper we prove a new estimate for the triple correlations associated to this problem, establishing an asymptotic expression for the third moment of the number of elements of $S_{α,N}$ in a random interval of length $L/N$, provided that $L > N^{1/4+\varepsilon}$. The threshold of $1/4$ is substantially smaller than the threshold of $1/2$ (which is the threshold that would be given by a naïve discrepancy estimate). Unlike the theory of pair correlations, rather little is known about triple correlations of the dilations $(αa_n \, \text{mod } 1)_{n=1}^{\infty} $ for a non-lacunary sequence $(a_n)_{n=1}^{\infty} $ of increasing integers. This is partially due to the fact that second moment of the triple correlation function is difficult to control, and thus standard techniques involving variance bounds are not applicable. We circumvent this impasse by using an argument inspired by works of Rudnick--Sarnak--Zaharescu and Heath-Brown, which connects the triple correlation function to some modular counting problems. In an appendix we comment on the relationship between discrepancy and correlation functions, answering a question of Steinerberger.
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Gap statistics and higher correlations for geometric progressions modulo one
Koksma's equidistribution theorem from 1935 states that for Lebesgue almost every $α>1$, the fractional parts of the geometric progression $(α^{n})_{n\geq1}$ are equidistributed modulo one. In the present paper we sharpen this result by showing that for almost every $α>1$, the correlations of all finite orders and hence the normalized gaps of $(α^{n})_{n\geq1}$ mod 1 have a Poissonian limit distribution, thereby resolving a conjecture of the two first named authors. While an earlier approach used probabilistic methods in the form of martingale approximation, our reasoning in the present paper is of an analytic nature and based upon the estimation of oscillatory integrals. This method is robust enough to allow us to extend our results to a natural class of sub-lacunary sequences.
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On the correlations of $n^α$ mod 1
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 $α>0$ is not an integer. For sharpening this knowledge to local statistics, the $k$-level correlation functions of the sequence $(\{n^α\})_{n\geq1}$ are of fundamental importance. We prove that for each $k\ge2,$ the $k$-level correlation function $R_k$ is Poissonian for almost every $α>4k^2-4k-1$.
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The metric theory of the pair correlation function of real-valued lacunary sequences
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 $αa(x)$ is Poissonian for Lebesgue almost every $α\in \mathbb{R}$. By using harmonic analysis, our result - irrespective of the choice of the real-valued sequence $\{ a(x) \}_{x=1}^{\infty}$ - can essentially be reduced to showing that the number of solutions to the Diophantine inequality $$ \vert n_1 (a(x_1)-a(y_1))- n_2(a(x_2)-a(y_2)) \vert < 1 $$ in integer six-tuples $(n_1,n_2,x_1,x_2,y_1,y_2)$ located in the box $[-N,N]^6$ with the ``excluded diagonals'', that is $$x_1\neq y_1, \quad x_2 \neq y_2, \quad (n_1,n_2)\neq (0,0),$$ is at most $N^{4-δ}$ for some fixed $δ>0$, for all sufficiently large $N$.
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The Duffin-Schaeffer conjecture with extra divergence
The Duffin-Schaeffer conjecture is a fundamental unsolved problem in metric number theory. It asserts that for every non-negative function $ψ:~\mathbb{N} \rightarrow \mathbb{R}$ for almost all reals $x$ there are infinitely many coprime solutions $(a,n)$ to the inequality $|nx - a| < ψ(n)$, provided that the series $\sum_{n=1}^\infty ψ(n) φ(n) /n$ is divergent. In the present paper we prove that the conjecture is true under the "extra divergence" assumption that divergence of the series still holds when $ψ(n)$ is replaced by $ψ(n) / (\log n)^\varepsilon$ for some $\varepsilon > 0$. This improves a result of Beresnevich, Harman, Haynes and Velani, and solves a problem posed by Haynes, Pollington and Velani.
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The discrepancy of $(n_kx)$ with respect to certain probability measures
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{equation*} \frac{1}{4} \leq \limsup_{N\to\infty}\frac{N D_N(n_kx)}{\sqrt{N\log\log N}} \leq C \end{equation*} for some constant $C>0$, proving a conjecture of Haynes, Jensen and Kristensen. This allows a slight improvement on their previous result on products of the form $q\|qα\| \|qβ-γ\| $.
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The maximal order of iterated multiplicative functions
Following Wigert, various authors, including Ramanujan, Gronwall, Erdős, Ivić, Schwarz, Wirsing, and Shiu, determined the maximal order of several multiplicative functions, generalizing Wigert's result $$\max_{n\leq x} \log d(n) = \frac{\log x}{\log \log x} (\log 2 + o(1)). $$ On the contrary, for many multiplicative functions, the maximal order of iterations of the functions remains widely open. The case of the iterated divisor function was only solved recently, answering a question of Ramanujan from 1915. Here we determine the maximal order of $\log f(f(n))$ for a class of multiplicative functions $f$. In particular, this class contains functions counting ideals of given norm in the ring of integers of an arbitrary, fixed quadratic number field. As a consequence, we determine such maximal orders for several multiplicative $f$ arising as a normalized function counting representations by certain binary quadratic forms. Incidentally, for the non-multiplicative function $r_2$ which counts how often a positive integer is represented as a sum of two squares, this entails the asymptotic formula $$ \max_{n\leq x} \log r_2(r_2(n))= \frac{\sqrt{\log x}}{\log \log x} (c/\sqrt{2}+o(1))$$ with some explicitly given constant $c>0$.
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There is no Khintchine threshold for metric pair correlations
We consider sequences of the form $\left(a_{n} α\right)_{n}$ mod 1, where $α\in\left[0,1\right]$ and where $\left(a_{n}\right)_{n}$ is a strictly increasing sequence of positive integers. If the asymptotic distribution of the pair correlations of this sequence follows the Poissonian model for almost all $α$ in the sense of Lebesgue measure, we say that $(a_n)_n$ has the metric pair correlation property. Recent research has revealed a connection between the metric theory of pair correlations of such sequences, and the additive energy of truncations of $(a_n)_{n}$. Bloom, Chow, Gafni and Walker speculated that there might be a convergence/divergence criterion which fully characterises the metric pair correlation property in terms of the additive energy, similar to Khintchine's criterion in the metric theory of Diophantine approximation. In the present paper we give a negative answer to such speculations, by showing that such a criterion does not exist. To this end, we construct a sequence $(a_n)_n$ having large additive energy which, however, maintains the metric pair correlation property.
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Higher-rank Bohr sets and multiplicative diophantine approximation
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 posed by Beresnevich, Haynes and Velani in 2015. Hitherto, this was only known on the plane, as previous approaches relied heavily on the theory of continued fractions. Using reduced successive minima in lieu of continued fractions, we develop the structural theory of Bohr sets of arbitrary rank, in the context of diophantine approximation. In addition, we generalise the theory and result to the inhomogeneous setting. To deal with this inhomogeneity, we employ diophantine transference inequalities in lieu of the three distance theorem.
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On linear relations for Dirichlet series formed by recursive sequences of second order
Let $F_n$ and $L_n$ be the Fibonacci and Lucas numbers, respectively. Four corresponding zeta functions in $s$ are defined by \[ζ_F(s) \,:=\, \sum_{n=1}^{\infty} \frac{1}{F_n^s}\,,\quad ζ_F^*(s) \,:=\,\sum_{n=1}^{\infty} \frac{{(-1)}^{n+1}}{F_n^s}\,,\quad ζ_L(s) \,:=\, \sum_{n=1}^{\infty} \frac{1}{L_n^s}\,,\quad ζ_L^*(s) \,:=\, \sum_{n=1}^{\infty} \frac{{(-1)}^{n+1}}{L_n^s} \,.\] For positive integers $s$ the transcendence of these values is known as well as algebraic independence or dependence results for sets of these numbers. In this paper, we investigate linear forms in the above zeta functions and determine the dimension of linear spaces spanned by such linear forms. In particular, it is established that for any positive integer $m$ the solutions of \[\sum_{s=1}^m \big( \,t_sζ_F(2s) + u_sζ_F^*(2s) + v_sζ_L(2s) + w_sζ_L^*(2s) \,\big) \,=\, 0 \] with $t_{s},u_{s},v_{s},w_{s} \in \mathbb{Q}$ $(1\leq s\leq m)$ form a $\mathbb{Q}$-vector space of dimension $m$. This proves a conjecture from the Ph.D. thesis of M. Stein, who, in 2012, was inspired by the relation $-2ζ_F(2)+ζ_F^*(2)+5ζ_L^*(2)=0$. All the results are also true for zeta functions in $s$ where the Fibonacci and Lucas numbers are replaced by numbers from sequences satisfying a second order recurrence formula.
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On Exceptional Sets in the Metric Poissonian Pair Correlations problem
Let $\left(a_{n}\right)_{n}$ be a strictly increasing sequence of positive integers, denote by $A_{N}=\left\{ a_{n}:\,n\leq N\right\} $ its truncations, and let $α\in\left[0,1\right]$. We prove that if the additive energy $E\left(A_{N}\right)$ of $A_{N}$ is in $Ω\left(N^{3}\right)$, then the sequence $\left(\left\langle αa_{n}\right\rangle \right)_{n}$ of fractional parts of $αa_{n}$ does not have Poissonian pair correlations (PPC) for almost every $α$ in the sense of Lebesgue measure. Conversely, it is known that $E\left(A_{N}\right)=\mathcal{O}\left(N^{3-\varepsilon}\right)$, for some fixed $\varepsilon>0$, implies that $\left(\left\langle αa_{n}\right\rangle \right)_{n}$ has PPC for almost every $α$. This note makes a contribution to investigating the energy threshold for $E\left(A_{N}\right)$ to imply this metric distribution property. We establish, in particular, that there exist sequences $\left(a_{n}\right)_{n}$ with \[ E\left(A_{N}\right)=Θ\left(\frac{N^{3}}{\log\left(N\right)\log\left(\log N\right)}\right) \] such that the set of $α$ for which $\left(αa_{n}\right)_{n}$ does not have PPC is of full Lebesgue measure. Moreover, we show that for any fixed $\varepsilon>0$ there are sequences $\left(a_{n}\right)_{n}$ with $E\left(A_{N}\right)=Θ\left(\frac{N^{3}}{\log\left(N\right)\left(\log\log N\right)^{1+\varepsilon}}\right)$ satisfying that the set of $α$ for which the sequence $\left(\bigl\langleαa_{n}\bigr\rangle\right)_{n}$ does not have PPC is of full Hausdorff dimension.
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On a Counting Theorem of Skriganov
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 terms of a certain function $ν(Γ^\perp,\cdot)$ of the dual lattice $Γ^\perp$, and we carefully analyse the relation of this quantity with $ν(Γ,\cdot)$. In particular, we show that $ν(Γ^\perp,\cdot)=ν(Γ,\cdot)$ for any unimodular lattice of rank 2, but that for higher ranks it is in general not possible to bound one function in terms of the other. Finally, we apply our counting theorem to establish asymptotics for the number of Diophantine approximations with bounded denominator as the denominator bound gets large.
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On the regularity of primes in arithmetic progressions
We prove that for a positive integer $k$ the primes in certain kinds of intervals can not distribute too 'uniformly' among the reduced residue classes modulo $k$. Hereby, we prove a generalization of a conjecture of Recaman and establish our results in a much more general situation, in particular for prime ideals in number fields.
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