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Christoph Aistleitner

Publications and source records attributed to Christoph Aistleitner.

At least 19 recordsLinked to original sources
arXiv2025

Arithmetic sensitivity of cumulant growth in lacunary sums: transcendental versus algebraic ratio limits

We study the asymptotic behavior of cumulants of lacunary trigonometric sums $S_n(ω) := \sum_{k=1}^n \cos (2 πa_k ω)$, $ω\in[0,1]$, and show that cumulant growth is highly sensitive to the arithmetic structure of the sequence $(a_k)_{k \geq 1}$ of positive integers. In particular, if $\lim_{k \to \infty} a_{k+1}/a_k = η> 1$ for some transcendental number $η$, we prove that for every $m\in \mathbb N$ the $m$-th cumulant of $S_n$ is asymptotically equivalent to the $m$-th cumulant of the ``independent model'' $\widetilde{S}_n := \sum_{k=1}^n \cos (2 πa_k U_k)$, where $U_1, U_2, \dots$ are independent random variables having uniform distribution on $[0,1]$. In particular, the order of growth of the cumulants as $n \to \infty$ is linear in this case. We also show that the transcendence condition for $\lim_{k \to \infty} a_{k+1}/a_k$ is in general necessary: when the ratio limit $η$ is algebraic, the cumulants of $S_n$ may have a different asymptotic order from those of $\widetilde{S}_n$. For instance, for $a_k = 2^k+1$ (with $η= 2$), the sixth cumulant of $S_n$ grows quadratically in $n$. In contrast, for $a_k = 2^k$ (again $η= 2$) or when $(a_k)_{k \geq 1}$ is the Fibonacci sequence (with $η= (1+\sqrt 5)/2$), the $m$-th cumulant of $S_n$ grows linearly as $n\to\infty$, but with a growth rate that differs from the one of the independent model $\widetilde{S}_n$. Overall, our results show that the asymptotic behavior of the cumulants of lacunary trigonometric sums depends on arithmetic effects in a very delicate way. This is particularly remarkable since many other probabilistic limit theorems, such as the Central Limit Theorem, hold for lacunary trigonometric sums in a universal way without any such sensitivity towards arithmetic effects.

math.NT↗
arXiv2025

A sharp threshold for arithmetic effects on the tail probabilities of lacunary sums

A classical observation in analysis asserts that lacunary systems of dilated functions show many properties which are also typical for systems of independent random variables. For example, if $(n_k)_{k \ge 1}$ is a sequence of integers satisfying the Hadamard gap condition $n_{k+1}/n_k\ge q > 1,~k \ge 1$, then the normalized sums $\sum_{n=1}^N \cos(2πn_k x)$, considered on the probability space $[0,1]$ with Borel $σ$-field and Lebesgue measure, satisfy the central limit theorem (CLT) and the law of the iterated logarithm (LIL). Remarkably, the situation becomes much more deliacate when the trigonometric function $\cos(2 πx)$ is replaced by a more general 1-periodic function $f$, and fine arithmetic properties of the sequence $(n_k)_{k \ge 1}$ come into play. The most relevant arithmetic property can be phrased in terms of the number of solutions of certain 2-variable Diophantine equations. Recently, the authors proved that the validity of the LIL requires a strictly stronger Diophantine criterion than the CLT. In the present paper we show that this is only a special case of a wide-ranging general principle: there is a sharp cutoff, which can be expressed in form of a Diophantine criterion on the sequence $(n_k)_{k \ge 1}$, at which the tail probabilities of $\sum_{k=1}^N f(n_k x)$ change from Gaussian to potentially erratic behavior. More precisely, let $L(N,a,b,c)$ be the number of solutions $(k,\ell)$ of the equation $a n_k - b n_\ell= c$, where $1\leq k,\ell \leq N$. Roughly speaking, we prove: if $L(N,a,b,c) \le N / g_N$ for some $g_N$, then $\mathbb{P} \left[\sum_{k=1}^N f(n_k x) > t \|f\|_2 \sqrt{N} \right]$ is asymptotically is accordance with standard normal behavior for all $t$ up to $\sqrt{2 \log g_N}$. We also show that this criterion is optimal in the sense that under the same premises, the conclusion can fail to be true for values of $t$ beyond this threshold.

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arXiv2025

On the behavior of the colored Jones polynomial of the figure-eight knot under modular transformations

The colored Jones polynomial $J_{K,N}$ is an important quantum knot invariant in low-dimensional topology. In his seminal paper on quantum modular forms, Zagier predicted the behavior of $J_{K,0}(e^{2 πi x})$ under the action of $SL_2(\mathbb{Z})$ on $x \in \mathbb{Q}$. More precisely, Zagier made a prediction on the asymptotic value of the quotient $J_{K,0}(e^{2 πi γ(x)})/ J_{K,0}(e^{2 πi x})$ for fixed $γ\in SL_2(\mathbb{Z})$, as $x \to \infty$ along rationals with bounded denominator. In the case of the figure-eight knot $4_1$, which is the most accessible case, there is an explicit formula for $J_{4_1,0}(e^{2 πi x})$ as a sum of certain trigonometric products called Sudler products. By periodicity, the behavior of $J_{4_1,0}(e^{2 πi x})$ under the mapping $x \mapsto x+1$ is trivial. For the second generator of $SL_2(\mathbb{Z})$, Zagier conjectured that with respect to the mapping $x \mapsto 1/x$, the quotient $h(x) = \log ( J_{4_1,0}(e^{2 πi x}) / J_{4_1,0}(e^{2 πi /x}))$ can be extended to a function on $\mathbb{R}$ that is continuous at all irrationals. This conjecture was recently established by Aistleitner and Borda in the case of all irrationals that have an unbounded sequence of partial quotients in their continued fraction expansion. In the present paper we prove Zagier's continuity conjecture in full generality.

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arXiv2025

Moment generating functions and moderate deviation principles for lacunary trigonometric sums

In a recent paper, Aistleitner, Gantert, Kabluchko, Prochno and Ramanan studied large deviation principles (LDPs) for lacunary trigonometric sums $\sum_{n=1}^N \cos(2 πn_k x)$, where the sequence $(n_k)_{k \geq 1}$ satisfies the Hadamard gap condition $n_{k+1} / n_k \geq q > 1$ for $k \geq 1$. A crucial ingredient in their work were asymptotic estimates for the moment generating function (MGF) of such sums, which turned out to depend on the fine arithmetic structure of the sequence $(n_k)_{k \geq 1}$ in an intricate way. In the present paper we carry out a detailed study of the MGF for lacunary trigonometric sums (without any structural assumptions on the underlying sequence, other than lacunarity), and we determine the sharp threshold where arithmetic effects start to play a role. As an application, we prove moderate deviation principles for lacunary trigonometric sums, and show that the tail probabilities are in accordance with Gaussian behavior throughout the whole range between the central limit theorem and the LDP regime.

math.PR↗
arXiv2025

Almost sure asymptotics for the number variance of dilations of integer sequences

Let $(x_n)_{n=1}^\infty$ be a sequence of integers. We study the number variance of dilations $(αx_n)_{n=1}^\infty$ modulo 1 in intervals of length $S$, and establish pseudorandom (Poissonian) behavior for Lebesgue-almost all $α$ throughout a large range of $S$, subject to certain regularity assumptions imposed upon $(x_n)_{n=1}^\infty$. For the important special case $x_n = p(n)$, where $p$ is a polynomial with integer coefficients of degree at least 2, we prove that the number variance is Poissonian for almost all $α$ throughout the range $0 \leq S \leq (\log N)^{-c}$, for a suitable absolute constant $c>0$. For more general sequences $(x_n)_{n=1}^\infty$, we give a criterion for Poissonian behavior for generic $α$ which is formulated in terms of the additive energy of the finite truncations $(x_n)_{n=1}^N$.

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arXiv2024

A conjecture of Zagier and the value distribution of quantum modular forms

In his influential paper on quantum modular forms, Zagier developed a conjectural framework describing the behavior of certain quantum knot invariants under the action of the modular group on their arguments. More precisely, when $J_{K,0}$ denotes the colored Jones polynomial of a knot $K$, Zagier's modularity conjecture describes the asymptotics of the quotient $J_{K,0} (e^{2 πi γ(x)}) / J_{K,0}(e^{2 πi x})$ as $x \to \infty$ along rationals with bounded denominators, where $γ\in \mathrm{SL}(2,\mathbb{Z})$. This problem is most accessible for the figure-eight knot $4_1$, where the colored Jones polynomial has a simple explicit expression in terms of the $q$-Pochhammer symbol. Zagier also conjectured that the function $h(x) = \log (J_{4_1,0} (e^{2 πi x}) / J_{4_1,0}(e^{2 πi /x}))$ can be extended to a function on $\mathbb{R}$ which is continuous at irrationals. In the present paper, we prove Zagier's continuity conjecture for all irrationals for which the sequence of partial quotients in the continued fraction expansion is unbounded. In particular, the continuity conjecture holds almost everywhere on the real line. We also establish a smooth approximation of $h$, uniform over all rationals, in accordance with the modularity conjecture. As an application, we find the limit distribution (after a suitable centering and rescaling) of $\log J_{4_1,0}(e^{2 πi x})$, when $x$ ranges over all reduced rationals in $(0,1)$ with denominator at most $N$, as $N \to \infty$, thereby confirming a conjecture of Bettin and Drappeau.

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arXiv2024

Lacunary sequences in analysis, probability and number theory

In this paper we present the theory of lacunary trigonometric sums and lacunary sums of dilated functions, from the origins of the subject up to recent developments. We describe the connections with mathematical topics such as equidistribution and discrepancy, metric number theory, normality, pseudorandomness, Diophantine equations, and the subsequence principle. In the final section of the paper we prove new results which provide necessary and sufficient conditions for the central limit theorem for subsequences, in the spirit of Nikishin's resonance theorem for convergence systems. More precisely, we characterize those sequences of random variables which allow to extract a subsequence satisfying a strong form of the central limit theorem.

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arXiv2023

Diophantine conditions in the law of the iterated logarithm for lacunary systems

It is a classical observation that lacunary function systems exhibit many properties which are typical for systems of independent random variables. However, it had already been observed by Erdős and Fortet in the 1950s that probability theory's limit theorems may fail for lacunary sums $\sum f(n_k x)$ if the sequence $(n_k)_{k \geq 1}$ has a strong arithmetic ''structure''. The presence of such structure can be assessed in terms of the number of solutions $k,\ell$ of two-term linear Diophantine equations $a n_k - b n_\ell = c$. As the first author proved with Berkes in 2010, saving an (arbitrarily small) unbounded factor for the number of solutions of such equations compared to the trivial upper bound, rules out pathological situations as in the Erdős--Fortet example, and guarantees that $\sum f(n_k x)$ satisfies the central limit theorem (CLT) in a form which is in accordance with true independence. In contrast, as shown by the first author, for the law of the iterated logarithm (LIL) the Diophantine condition which suffices to ensure ''truly independent'' behavior requires saving this factor of logarithmic order. In the present paper we show that, rather surprisingly, saving such a logarithmic factor is actually the optimal condition in the LIL case. This result reveals the remarkable fact that the arithmetic condition required of $(n_k)_{k \geq 1}$ to ensure that $\sum f(n_k x)$ shows ''truly random'' behavior is a different one at the level of the CLT than it is at the level of the LIL: the LIL requires a stronger arithmetic condition than the CLT does.

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arXiv2023

On the distribution of partial quotients of reduced fractions with fixed denominator

In this paper, we study distributional properties of the sequence of partial quotients in the continued fraction expansion of fractions $a/N$, where $N$ is fixed and $a$ runs through the set of mod $N$ residue classes which are coprime with $N$. Our methods cover statistics such as the sum of partial quotients, the maximal partial quotient, the empirical distribution of partial quotients, Dedekind sums, and much more. We prove a sharp concentration inequality for the sum of partial quotients, and sharp tail estimates for the maximal partial quotient and for Dedekind sums, all matching the tail behavior in the limit laws which are known under an extra averaging over the set of possible denominators $N$. We show that the distribution of partial quotients of reduced fractions with fixed denominator gives a very good fit to the Gauss-Kuzmin distribution. As corollaries we establish the existence of reduced fractions with a small sum of partial quotients resp. a small maximal partial quotient.

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arXiv2022

On the metric theory of approximations by reduced fractions: a quantitative Koukoulopoulos-Maynard theorem

Let $ψ: \mathbb{N} \to [0,1/2]$ be given. The Duffin-Schaeffer conjecture, recently resolved by Koukoulopoulos and Maynard, asserts that for almost all reals $α$ there are infinitely many coprime solutions $(p,q)$ to the inequality $|α- p/q| < ψ(q)/q$, provided that the series $\sum_{q=1}^\infty φ(q) ψ(q) / q$ is divergent. In the present paper, we establish a quantitative version of this result, by showing that for almost all $α$ the number of coprime solutions $(p,q)$, subject to $q \leq Q$, is of asymptotic order $\sum_{q=1}^Q 2 φ(q) ψ(q) / q$. The proof relies on the method of GCD graphs as invented by Koukoulopoulos and Maynard, together with a refined overlap estimate coming from sieve theory, and number-theoretic input on the "anatomy of integers". The key phenomenon is that the system of approximation sets exhibits "asymptotic independence on average" as the total mass of the set system increases.

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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.

math.NT↗
arXiv2021

Difference sets and the metric theory of small gaps

Let $(a_n)_{n \geq 1}$ be a sequence of distinct positive integers. In a recent paper Rudnick established asymptotic upper bounds for the minimal gaps of $\{a_n α\bmod 1, 1 \leq n \leq N\}$ as $N \to \infty$, valid for Lebesgue-almost all $α$ and formulated in terms of the additive energy of $\{a_1, \dots, a_N\}$. In the present paper we argue that the metric theory of minimal gaps of such sequences is not controlled by the additive energy, but rather by the cardinality of the difference set of $\{a_1, \dots, a_N\}$. We establish a (complicated) sharp convergence/divergence test for the typical asymptotic order of the minimal gap, and prove (slightly weaker) general upper and lower bounds which allow for a direct application. A major input for these results comes from the recent proof of the Duffin--Schaeffer conjecture by Koukoulopoulos and Maynard. We show that our methods give very precise results for slowly growing sequences whose difference set has relatively high density, such as the primes or the squares. Furthermore, we improve a metric result of Blomer, Bourgain, Rudnick and Radziwill on the order of the minimal gap in the eigenvalue spectrum of a rectangular billiard.

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arXiv2021

Quantum invariants of hyperbolic knots and extreme values of trigonometric products

In this paper we study the relation between the function $J_{4_1,0}$, which arises from a quantum invariant of the figure-eight knot, and Sudler's trigonometric product. We find $J_{4_1,0}$ up to a constant factor along continued fraction convergents to a quadratic irrational, and we show that its asymptotics deviates from the universal limiting behavior that has been found by Bettin and Drappeau in the case of large partial quotients. We relate the value of $J_{4_1,0}$ to that of Sudler's trigonometric product, and establish asymptotic upper and lower bounds for such Sudler products in response to a question of Lubinsky.

math.NT↗
arXiv2021

Maximizing Sudler products via Ostrowski expansions and cotangent sums

There is an extensive literature on the asymptotic order of Sudler's trigonometric product $P_N (α) = \prod_{n=1}^N |2 \sin (πn α)|$ for fixed or for "typical" values of $α$. In the present paper we establish a structural result, which for a given $α$ characterizes those $N$ for which $P_N(α)$ attains particularly large values. This characterization relies on the coefficients of $N$ in its Ostrowski expansion with respect to $α$, and allows us to obtain very precise estimates for $\max_{1 \le N \leq M} P_N(α)$ and for $\sum_{N=1}^M P_N(α)^c$ in terms of $M$, for any $c>0$. Furthermore, our arguments give a natural explanation of the fact that the value of the hyperbolic volume of the complement of the figure-eight knot appears generically in results on the asymptotic order of the Sudler product and of the Kashaev invariant.

math.NT↗
arXiv2021

A pair correlation problem, and counting lattice points with the zeta function

The pair correlation is a localized statistic for sequences in the unit interval. Pseudo-random behavior with respect to this statistic is called Poissonian behavior. The metric theory of pair correlations of sequences of the form $(a_n α)_{n \geq 1}$ has been pioneered by Rudnick, Sarnak and Zaharescu. Here $α$ is a real parameter, and $(a_n)_{n \geq 1}$ is an integer sequence, often of arithmetic origin. Recently, a general framework was developed which gives criteria for Poissonian pair correlation of such sequences for almost every real number $α$, in terms of the additive energy of the integer sequence $(a_n)_{n \geq 1}$. In the present paper we develop a similar framework for the case when $(a_n)_{n \geq 1}$ is a sequence of reals rather than integers, thereby pursuing a line of research which was recently initiated by Rudnick and Technau. As an application of our method, we prove that for every real number $θ>1$, the sequence $(n^θα)_{n \geq 1}$ has Poissonian pair correlation for almost all $α\in \mathbb{R}$.

math.NT↗
arXiv2020

Large Deviation Principles for Lacunary Sums

Let $(a_k)_{k\in\mathbb N}$ be a sequence of integers satisfying the Hadamard gap condition $a_{k+1}/a_k>q>1$ for all $k\in\mathbb N$, and let $$ S_n(ω) = \sum_{k=1}^n\cos(2πa_k ω),\qquad n\in\mathbb N,\;ω\in [0,1]. $$ The lacunary trigonometric sum $S_n$ is known to exhibit several properties typical for sums of independent random variables. In this paper we initiate the investigation of large deviation principles (LDPs) for $S_n$. Under the large gap condition $a_{k+1}/a_k\to\infty$, we prove that $(S_n/n)_{n\in\mathbb N}$ satisfies an LDP with speed $n$ and the same rate function $\tilde{I}$ as for sums of independent random variables with the arcsine distribution, but show that the LDP may fail to hold when we only assume the Hadamard gap condition. However, we prove that in the special case $a_k=q^k$ for some $q\in \{2,3,\ldots\}$, $(S_n/n)_{n\in\mathbb N}$ satisfies an LDP with speed $n$ and a rate function $I_q$ different from $\tilde{I}$. We also show that $I_q$ converges pointwise to $\tilde I$ as $q\to\infty$ and construct a random perturbation $(a_k)_{k\in\mathbb N}$ of the sequence $(2^k)_{k\in\mathbb N}$ for which $a_{k+1}/a_k\to 2$ as $k\to\infty$, but for which $(S_n/n)_{n\in\mathbb N}$ satisfies an LDP with the rate function $\tilde{I}$ as in the independent case and not, as one might na{ï}vely expect, with rate function $I_2$. We relate this fact to the number of solutions of certain Diophantine equations. Our results show that LDPs for lacunary trigonometric sums are sensitive to the arithmetic properties of $(a_k)_{k\in\mathbb N}$. This is particularly noteworthy since no such arithmetic effects are visible in the central limit theorem by Salem and Zygmund or in the law of the iterated logarithm by Erdös and Gál. Our proofs use a combination of tools from probability theory, harmonic analysis, and dynamical systems.

math.PR↗
arXiv2020

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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