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Jeffrey D. Vaaler

Publications and source records attributed to Jeffrey D. Vaaler.

At least 19 recordsLinked to original sources
arXiv2025

Effective equidistribution of norm one elements in CM-fields

For a number field $K$ let $\mathcal{S}_K$ be the maximal subgroup of the multiplicative group $K^\times$ that embeds into the unit circle under each embedding of $K$ into the complex numbers. The group $\mathcal{S}_K$ can be seen as an archimedean counterpart to the group of units $\mathcal{O}_K^\times$ of the ring of integers $\mathcal{O}_K$. If $K=\mathbb{Q}(\mathcal{S}_K)$ is a CM-field then $\mathcal{S}_K/{\mathop{\rm Tor}\nolimits}(K^\times)$ is a free abelian group of infinite rank. If $K=\mathbb{Q}(\mathcal{S}_K)$ is not a CM-field then $\mathcal{S}_K=\{\pm 1\}$. In the former case $\mathcal{S}_K$ is the kernel of the relative norm map from $K^\times$ to the multiplicative subgroup $k^\times$ of the maximal totally real subfield $k$ of $K$.

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arXiv2023

A bound for the exterior product of $S$-units

We generalize an inequality for the determinant of a real matrix proved by A. Schinzel, to more general exterior products of vectors in Euclidean space. We apply this inequality to the logarithmic embedding of $S$-units contained in a number field $k$. This leads to a bound for the exterior product of $S$-units expressed as a product of heights. Using a volume formula of P. McMullen we show that our inequality is sharp up to a constant that depends only on the rank of the $S$-unit group but not on the field $k$. Our inequality is related to a conjecture of F. Rodriguez Villegas.

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arXiv2022

Lower Bounds for Regulators of Number Fields in terms of their Discriminants

We prove inequalities that compare the regulator of a number field with its absolute discriminant. We refine some ideas in Silverman's work in 1984 where such general inequalities are first proven. In order to prove our main theorems, we combine these refinements with the authors' recent results on bounding the product of heights of relative units in a number field extension.

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arXiv2018

Lower bounds for Mahler measure that depend on the number of monomials

We prove a new lower bound for the Mahler measure of a polynomial in one and in several variables that depends on the complex coefficients, and the number of monomials. In one variable our result generalizes a classical inequality of Mahler. In $M$ variables our result depends on $\mathbb{Z}^M$ as an ordered group, and in general our lower bound depends on the choice of ordering.

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arXiv2018

A Dirichlet approximation theorem for group actions

If $G$ is a compact group acting continuously on a compact metric space $(X, m)$, we prove two results that generalize Dirichlet's classical theorem on Diophantine approximation. If $G$ is a noncommutative compact group of isometries, we obtain a noncommutative form of Dirichlet's theorem. We apply our general result to the special case of the unitary group $U(N)$ acting on the complex unit sphere, and obtain a noncommutative result in this setting.

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arXiv2018

On the height of solutions to norm form equations

Let $k$ be a number field. We consider norm form equations associated to a full $O_k$-module contained in a finite extension field $l$. It is known that the set of solutions is naturally a union of disjoint equivalence classes of solutions. We prove that each nonempty equivalence class of solutions contains a representative with Weil height bounded by an expression that depends on parameters defining the norm form equation.

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arXiv2017

Multiplicative approximation by the Weil height

Let $K/\mathbb{Q}$ be an algebraic extension of fields, and let $α\not= 0$ be contained in an algebraic closure of $K$. If $α$ can be approximated by roots of numbers in $K^{\times}$ with respect to the Weil height, we prove that some nonzero integer power of $α$ must belong to $K^{\times}$. More generally, let $K_1, K_2, \dots , K_N$, be algebraic extensions of $\mathb{Q}$ such that each pair of extensions includes one which is a (possibly infinite) Galois extension of a common subfield. If $α\not= 0$ can be approximated by a product of roots of numbers from each $K_n$ with respect to the Weil height, we prove that some nonzero integer power of $α$ must belong to the multiplicative group $K_1^{\times} K_2^{\times} \cdots K_N^{\times}$. Our proof of the more general result uses methods from functional analysis.

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arXiv2017

Minkowski's theorem on independent conjugate units

We call a unit $β$ in a Galois extension $l/\mathbb{Q}$ a Minkowski unit if the subgroup generated by $β$ and its conjugates over $\mathbb{Q}$ has maximum rank in the unit group of $l$. Minkowski showed the existence of such units in every Galois extension. We will give a new proof to Minkowski's theorem and show that there exists a Minkowski unit $β\in l$ such that the Weil height of $β$ is comparable with the sum of the heights of a fundamental system of units of $l$. Our proof implies a bound on the index of the subgroup generated by the algebraic conjugates of $β$ in the unit group of $l$. If $k$ is an intermediate field such that \begin{equation*} \mathbb{Q} \subseteq k \subseteq l, \end{equation*} and $l/\mathbb{Q}$ and $k/\mathbb{Q}$ are Galois extensions, we prove an analogous bound for the subgroup of relative units.

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arXiv2016

Heights, Regulators and Schinzel's determinant inequality

We prove inequalities that compare the size of an S-regulator with a product of heights of multiplicatively independent S-units. Our upper bound for the S-regulator follows from a general upper bound for the determinant of a real matrix proved by Schinzel. The lower bound for the S-regulator follows from Minkowski's theorem on successive minima and a volume formula proved by Meyer and Pajor. We establish similar upper bounds for the relative regulator of an extension of number fields.

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arXiv2014

Number fields without small generators

Let $D>1$ be an integer, and let $b=b(D)>1$ be its smallest divisor. We show that there are infinitely many number fields of degree $D$ whose primitive elements all have relatively large height in terms of $b$, $D$ and the discriminant of the number field. This provides a negative answer to a questions of W. Ruppert from 1998 in the case when $D$ is composite. Conditional on a very weak form of a folk conjecture about the distribution of number fields, we negatively answer Ruppert's question for all $D>3$.

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arXiv2013

Sums of products of fractional parts

We prove upper and lower bounds for certain sums of products of fractional parts by using majoring and minorizing functions from Fourier analysis. In special cases the upper bounds are sharp if there exist counterexamples to the Littlewood conjecture in Diophantine approximation. We introduce a generalization of such counterexamples which we call strongly badly approximable matrices. And we prove a transference principle for strongly badly approximable matrices.

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arXiv2012

Heights on Groups and Small Multiplicative Dependencies

We generalize the absolute logarithmic Weil height from elements of the multiplicative group of algebraic numbers modulo torsion, to finitely generated subgoups. The height of a finitely generated subgroup is shown to equal the volume of a certain naturally occurring, convex, symmetric subset of Euclidean space. This connection leads to a bound on the norm of integer vectors that give multiplicative dependencies among finite sets of algebraic numbers.

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arXiv2012

A note on generators of number fields

We establish upper bounds for the smallest height of a generator of a number field $k$ over the rational field $\Q$. Our first bound applies to all number fields $k$ having at least one real embedding. We also give a second conditional result for all number fields $k$ such that the Dedekind zeta-function associated to the Galois closure of $k/\Q$ satisfies GRH. This provides a partial answer to a question of W. Ruppert.

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arXiv2011

Gaussian Subordination for the Beurling-Selberg Extremal Problem

We determine extremal entire functions for the problem of majorizing, minorizing, and approximating the Gaussian function $e^{-πλx^2}$ by entire functions of exponential type. This leads to the solution of analogous extremal problems for a wide class of even functions that includes most of the previously known examples (for instance \cite{CV2}, \cite{CV3}, \cite{GV} and \cite{Lit}), plus a variety of new interesting functions such as $|x|^α$ for $-1 < α$; \,$\log \,\bigl((x^2 + α^2)/(x^2 + β^2)\bigr)$, for $0 \leq α< β$;\, $\log\bigl(x^2 + α^2\bigr)$; and $x^{2n} \log x^2$\,, for $n \in \N$. Further applications to number theory include optimal approximations of theta functions by trigonometric polynomials and optimal bounds for certain Hilbert-type inequalities related to the discrete Hardy-Littlewood-Sobolev inequality in dimension one.

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arXiv2009

Martingale differences and the metric theory of continued fractions

We investigate a collection of orthonormal functions that encodes information about the continued fraction expansion of real numbers. When suitably ordered these functions form a complete system of martingale differences and are a special case of a class of martingale differences considered by R. F. Gundy. By applying known results for martingales we obtain corresponding metric theorems for the continued fraction expansion of almost all real numbers.

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arXiv2008

Some extremal functions in Fourier analysis, III

We obtain the best approximation in $L^1(\R)$, by entire functions of exponential type, for a class of even functions that includes $e^{-λ|x|}$, where $λ>0$, $\log |x|$ and $|x|^α$, where $-1 < α< 1$. We also give periodic versions of these results where the approximating functions are trigonometric polynomials of bounded degree.

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arXiv2008

Some extremal functions in Fourier analysis, II

We obtain extremal majorants and minorants of exponential type for a class of even functions on $\R$ which includes $\log |x|$ and $|x|^α$, where $-1 < α< 1$. We also give periodic versions of these results in which the majorants and minorants are trigonometric polynomials of bounded degree. As applications we obtain optimal estimates for certain Hermitian forms, which include discrete analogues of the one dimensional Hardy-Littlewood-Sobolev inequalities. A further application provides an Erdös-Turán-type inequality that estimates the sup norm of algebraic polynomials on the unit disc in terms of power sums in the roots of the polynomials.

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