Nonzero Fourier Coefficients Research Articles

On the largest prime factor of non-zero Fourier coefficients of Hecke eigenforms

Abstract Let τ denote the Ramanujan tau function. One is interested in possible prime values of τ function. Since τ is multiplicative and τ ⁢ ( n ) {\tau(n)} is odd if and only if n is an odd square, we only need to consider τ ⁢ ( p 2 ⁢ n ) {\tau(p^{2n})} for primes p and natural numbers n ≥ 1 {n\geq 1} . This is a rather delicate question. In this direction, we show that for any ϵ > 0 {\epsilon>0} and integer n ≥ 1 {n\geq 1} , the largest prime factor of τ ⁢ ( p 2 ⁢ n ) {\tau(p^{2n})} , denoted by P ⁢ ( τ ⁢ ( p 2 ⁢ n ) ) {P(\tau(p^{2n}))} , satisfies P ⁢ ( τ ⁢ ( p 2 ⁢ n ) ) > ( log ⁡ p ) 1 8 ⁢ ( log ⁡ log ⁡ p ) 3 8 - ϵ P(\tau(p^{2n}))>(\log p)^{\frac{1}{8}}(\log\log p)^{\frac{3}{8}-\epsilon} for almost all primes p. This improves a recent work of Bennett, Gherga, Patel and Siksek. Our results are also valid for any non-CM normalized Hecke eigenforms with integer Fourier coefficients.

A generalization of a theorem of Rothschild and van Lint

A classical result of Rothschild and van Lint asserts that if every non-zero Fourier coefficient of a Boolean function f over F2n has the same absolute value, namely |fˆ(α)|=1/2k for every α in the Fourier support of f, then f must be the indicator function of some affine subspace of dimension n−k. In this paper we slightly generalize their result. Our main result shows that, roughly speaking, Boolean functions whose Fourier coefficients take values in the set {−2/2k,−1/2k,0,1/2k,2/2k} are indicator functions of two disjoint affine subspaces of dimension n−k or four disjoint affine subspaces of dimension n−k−1. Our main technical tools are results from additive combinatorics which offer tight bounds on the affine span size of a subset of F2n when the doubling constant of the subset is small.

Left-Invertibility of Rank-One Perturbations

For each isometry V acting on some Hilbert space and a pair of vectors f and g in the same Hilbert space, we associate a nonnegative number c(V; f, g) defined by $$\begin{aligned} c(V; f,g) = (\Vert f\Vert ^2 - \Vert V^*f\Vert ^2) \Vert g\Vert ^2 + |1 + \langle V^*f , g\rangle |^2. \end{aligned}$$We prove that the rank-one perturbation \(V + f \otimes g\) is left-invertible if and only if $$\begin{aligned} c(V;f,g) \ne 0. \end{aligned}$$We also consider examples of rank-one perturbations of isometries that are shift on some Hilbert space of analytic functions. Here, shift refers to the operator of multiplication by the coordinate function z. Finally, we examine \(D + f \otimes g\), where D is a diagonal operator with nonzero diagonal entries and f and g are vectors with nonzero Fourier coefficients. We prove that \(D + f\otimes g\) is left-invertible if and only if \(D+f\otimes g\) is invertible.

On the number of prime divisors and radicals of non-zero Fourier coefficients of Hilbert cusp forms

Abstract In this article, we derive lower bounds for the number of distinct prime divisors of families of non-zero Fourier coefficients of non-CM primitive cusp forms and more generally of non-CM primitive Hilbert cusp forms. In particular, for the Ramanujan Δ-function, we show that, for any ϵ > 0 \epsilon>0 , there exist infinitely many natural numbers 𝑛 such that τ ⁢ ( p n ) \tau(p^{n}) has at least 2 ( 1 - ϵ ) ⁢ log ⁡ n log ⁡ log ⁡ n 2^{(1-\epsilon)\frac{\log n}{\log\log n}} distinct prime factors for almost all primes 𝑝. This improves and refines the existing bounds. We also study lower bounds for absolute norms of radicals of non-zero Fourier coefficients of modular forms alluded to above.

Tractable circula densities from Fourier series

This article proposes an approach, based on infinite Fourier series, to constructing tractable densities for the bivariate circular analogues of copulas recently coined ‘circulas’. As examples of the general approach, we consider circula densities generated by various patterns of nonzero Fourier coefficients. The shape and sparsity of such arrangements are found to play a key role in determining the properties of the resultant models. The special cases of the circula densities we consider all have simple closed-form expressions involving no computationally demanding normalizing constants and display wide-ranging distributional shapes. A highly successful model identification tool and methods for parameter estimation and goodness-of-fit testing are provided for the circula densities themselves and the bivariate circular densities obtained from them using a marginal specification construction. The modelling capabilities of such bivariate circular densities are compared with those of five existing models in a numerical experiment, and their application illustrated in an analysis of wind directions.

Improved Bounds on Fourier Entropy and Min-entropy

Given a Boolean function f:{ -1,1} ^{n}→ { -1,1, define the Fourier distribution to be the distribution on subsets of [n], where each S ⊆ [n] is sampled with probability f ˆ (S) 2 . The Fourier Entropy-influence (FEI) conjecture of Friedgut and Kalai [28] seeks to relate two fundamental measures associated with the Fourier distribution: does there exist a universal constant C > 0 such that H(f ˆ2 ) ≤ C ⋅ Inf (f), where H (fˆ2) is the Shannon entropy of the Fourier distribution of f and Inf(f) is the total influence of f In this article, we present three new contributions toward the FEI conjecture: (1) Our first contribution shows that H(f ˆ2 ) ≤ 2 ⋅ aUC ⊕ (f), where aUC ⊕ (f) is the average unambiguous parity-certificate complexity of f . This improves upon several bounds shown by Chakraborty et al. [20]. We further improve this bound for unambiguous DNFs. We also discuss how our work makes Mansour's conjecture for DNFs a natural next step toward resolution of the FEI conjecture. (2) We next consider the weaker Fourier Min-entropy-influence (FMEI) conjecture posed by O'Donnell and others [50, 53], which asks if H ∞ fˆ2) ≤ C ⋅ Inf(f), where H ∞ fˆ2) is the min-entropy of the Fourier distribution. We show H ∞ (fˆ2) ≤ 2⋅C min ⊕ (f), where C min ⊕ (f) is the minimum parity-certificate complexity of f . We also show that for all ε≥0, we have H ∞ (fˆ2) ≤2 log⁡(∥f ˆ ∥1,ε/(1−ε)), where ∥f ˆ ∥1,ε is the approximate spectral norm of f . As a corollary, we verify the FMEI conjecture for the class of read- k DNFs (for constant k ). (3) Our third contribution is to better understand implications of the FEI conjecture for the structure of polynomials that 1/3-approximate a Boolean function on the Boolean cube. We pose a conjecture: no flat polynomial (whose non-zero Fourier coefficients have the same magnitude) of degree d and sparsity 2 ω(d) can 1/3-approximate a Boolean function. This conjecture is known to be true assuming FEI, and we prove the conjecture unconditionally (i.e., without assuming the FEI conjecture) for a class of polynomials. We discuss an intriguing connection between our conjecture and the constant for the Bohnenblust-Hille inequality, which has been extensively studied in functional analysis.

ON FUNDAMENTAL FOURIER COEFFICIENTS OF SIEGEL MODULAR FORMS

AbstractWe prove that if F is a nonzero (possibly noncuspidal) vector-valued Siegel modular form of any degree, then it has infinitely many nonzero Fourier coefficients which are indexed by half-integral matrices having odd, square-free (and thus fundamental) discriminant. The proof uses an induction argument in the setting of vector-valued modular forms. Further, as an application of a variant of our result and complementing the work of A. Pollack, we show how to obtain an unconditional proof of the functional equation of the spinor L-function of a holomorphic cuspidal Siegel eigenform of degree $3$ and level $1$ .

Determining Siegel modular forms of half-integral weight by their fundamental Fourier coefficients

We prove that a non-zero Siegel cusp form of half-integral weight and degree 2 on $\Gamma _0^{(2)}(4)$ has infinitely many non-zero Fourier coefficients indexed by semi-integral matrices having fundamental discriminant. This is the half-integral weight ve

A note on lower bounds of heights of non-zero Fourier coefficients of Hilbert cusp forms

It is a conjecture of Atkin and Serre that for any $$\epsilon > 0$$, there exists a constant $$c(\epsilon ) > 0$$ such that the Ramanujan $$\tau $$-function satisfies $$\begin{aligned} |\tau (p)| \ge c(\epsilon ) ~p^{9/2 - \epsilon } \end{aligned}$$for all sufficiently large primes. This in particular would settle Lehmer’s folklore conjecture for almost all primes and hence is widely open. In an interesting and elegant work, R. Murty, K. Murty, and T. Shorey showed that if $$\tau (n)$$ is odd, then $$|\tau (n)| > (\log n)^{\delta }$$ for some effective absolute constant $$\delta > 0$$. In this short note, building upon their work, we derive lower bounds for heights of certain Fourier coefficients of primitive Hilbert cusp forms. The extension to Hilbert modular forms brings in some new difficulties which are absent for the Ramanujan $$\tau $$-function and compel us to abandon absolute values and work with Weil heights. This brings in the further caveat that the bounds on prime powers no longer lead to bounds on arbitrary Fourier coefficients as the height inequalities are no longer compatible to derive such lower bounds. A major point in this circle of questions is the issue of non-vanishing of the eigenvalues $$c({\mathfrak {p}})$$ which are associated to the prime ideals $${\mathfrak {p}}$$ of the ambient number field. In the final section, we indicate a somewhat curious link between this and the oscillation of the real sequence $$\{c(\mathfrak {p^n})\}_{n\in {{\mathbb {N}}}}$$.

Alternation, sparsity and sensitivity: Bounds and exponential gaps

The well-known Sensitivity Conjecture regarding combinatorial complexity measures on Boolean functions states that for any Boolean function f:{0,1}n→{0,1}, block sensitivity of f is polynomially related to sensitivity of f (denoted by s(f)). From the complexity theory side, the Xor Log-Rank Conjecture states that for any Boolean function, f:{0,1}n→{0,1} the communication complexity of a related function f⊕:{0,1}n×{0,1}n→{0,1} (defined as f⊕(x,y)=f(x⊕y)) is bounded by polynomial in logarithm of the sparsity of f (the number of non-zero Fourier coefficients for f, denoted by sparsity(f)). Both the conjectures play a central role in the domains in which they are studied.A recent result of Lin and Zhang (2017) implies that to confirm the above two conjectures it suffices to upper bound alternation of f (denoted by alt(f)) for all Boolean functions f by polynomial in s(f) and logarithm of sparsity(f), respectively. In this context, we show the following results:•We show that there exists a family of Boolean functions for which alt(f) is at least exponential in s(f) and alt(f) is at least exponential in log⁡sparsity(f). En route to the proof, we also show an exponential gap between alt(f) and the decision tree complexity of f, which might be of independent interest.•As our main result, we show that, despite the above exponential gap between alt(f) and log⁡sparsity(f), the Xor Log-Rank Conjecture is true for functions with the alternation upper bounded by poly(log⁡n). It is easy to observe that the Sensitivity Conjecture is also true for this class of functions.•The starting point for the above result is the observation (derived from Lin and Zhang (2017)) that for any Boolean function f, deg(f)≤alt(f)deg2(f)degm(f) where deg(f), deg2(f) and degm(f) are the degrees of f over R, F2 and Zm respectively. We give three further applications of this bound: (1) We show that for Boolean functions f of constant alternation have deg2(f)=Ω(log⁡n). (2) Moreover, these functions also have high sparsity (exp⁡(Ω(deg(f))), thus partially answering a question of Kulkarni and Santha (2013). (3) We observe that our relation upper bounding real degree also improves the upper bound for influence to deg2(f)2⋅alt(f) improving Guo and Komargodski (2017).

On existence of generic cusp forms on semisimple algebraic groups

In this paper we discuss the existence of certain classes of cuspidal automorphic representations having non-zero Fourier coefficients for a general semisimple algebraic group $ G$ defined over a number field $ k$ such that its Archimedean group $ G_\infty $ is not compact. When $ G$ is quasi-split over $ k$, we obtain a result on existence of generic cuspidal automorphic representations which generalize results of Vigneras, Henniart, and Shahidi. We also discuss: (i) the existence of cuspidal automorphic forms with non-zero Fourier coefficients for congruence of subgroups of $ G_\infty $, and (ii) applications related to the work of Bushnell and Henniart on generalized Whittaker models.

The impact of ΛCDM substructure and baryon-dark matter transition on the image positions of quad galaxy lenses

The positions of multiple images in galaxy lenses are related to the galaxy mass distribution. Smooth elliptical mass profiles were previously shown to be inadequate in reproducing the quad population. In this paper, we explore the deviations from such smooth elliptical mass distributions. Unlike most other work, we use a model-free approach based on the relative polar image angles of quads, and their position in 3D space with respect to the fundamental surface of quads (FSQ). The FSQ is defined by quads produced by elliptical lenses. We have generated thousands of quads from synthetic populations of lenses with substructure consistent with Lambda cold dark matter (ΛCDM) simulations, and found that such perturbations are not sufficient to match the observed distribution of quads relative to the FSQ. The result is unchanged even when subhalo masses are increased by a factor of 10, and the most optimistic lensing selection bias is applied. We then produce quads from galaxies created using two components, representing baryons and dark matter. The transition from the mass being dominated by baryons in inner radii to being dominated by dark matter in outer radii can carry with it asymmetries, which would affect relative image angles. We run preliminary experiments using lenses with two elliptical mass components with non-identical axial ratios and position angles, perturbations from ellipticity in the form of non-zero Fourier coefficients a4 and a6, and artificially offset ellipse centres as a proxy for asymmetry at image radii. We show that combination of these effects is a promising way of accounting for quad population properties. We conclude that the quad population provides a unique and sensitive tool for constraining detailed mass distribution in the centres of galaxies.

On the gaps between nonzero Fourier coefficients of eigenforms with CM

Suppose [Formula: see text] is an elliptic curve over [Formula: see text] of conductor [Formula: see text] with complex multiplication (CM) by [Formula: see text], and [Formula: see text] is the corresponding cuspidal Hecke eigenform in [Formula: see text]. Then [Formula: see text]th Fourier coefficient of [Formula: see text] is nonzero in the short interval [Formula: see text] for all [Formula: see text] and for some [Formula: see text]. As a consequence, we produce infinitely many cuspidal CM eigenforms [Formula: see text] level [Formula: see text] and weight [Formula: see text] for which [Formula: see text] holds, for all [Formula: see text].

Non-vanishing of fundamental Fourier coefficients of paramodular forms

We prove that paramodular newforms of odd square-free level have infinitely many non-zero fundamental Fourier coefficients.

X-RAY THIRD-ORDER NONLINEAR RENNINGER EFFECT AND ROCKING CURVES

In the work third-order nonlinear Takagi’s equations for X-ray monochromatic waves are investigated for the forbidden reflection case. The forbidden dynamical diffraction in the nonlinear case is related to the presence in the nonlinear equations of the terms proportional to the zero order and the second order nonzero Fourier coefficients of the third-order nonlinear susceptibility. As a consequence, in the third-order nonlinear Bragg diffraction case, a nonlinear analogue of the well known Renninger effect takes place, which is considered theoretically and numerically. The numerical calculations show that in the Bragg geometry the nonlinear reflection curve’s behavior is the same as for not forbidden reflection, but for forbidden reflection the rocking curves are by several orders more sensitive to the input intensity value.

On the gaps between non-zero Fourier coefficients of cusp forms of higher weight

We show that if a modular cuspidal eigenform $f$ of weight $2k$ is $2$-adically close to an elliptic curve $E/\mathbb{Q}$, which has a cyclic rational $4$-isogeny, then $n$-th Fourier coefficient of $f$ is non-zero in the short interval $(X, X + cX^{\frac{1}{4}})$ for all $X \gg 0$ and for some $c > 0$. We use this fact to produce non-CM cuspidal eigenforms $f$ of level $N>1$ and weight $k > 2$ such that $i_f(n) \ll n^{\frac{1}{4}}$ for all $n \gg 0$.

Order parameter analysis for low-dimensional behaviors of coupled phase-oscillators.

Coupled phase-oscillators are important models related to synchronization. Recently, Ott-Antonsen(OA) ansatz is developed and used to get low-dimensional collective behaviors in coupled oscillator systems. In this paper, we develop a simple and concise approach based on equations of order parameters, namely, order parameter analysis, with which we point out that OA ansatz is rooted in the dynamical symmetry of order parameters. With our approach the scope of OA ansatz is identified as two conditions, i.e., the limit of infinitely many oscillators and only three nonzero Fourier coefficients of the coupling function. Coinciding with each of the conditions, a distinctive system out of the scope is taken into account and discussed with the order parameter analysis. Two approximation methods are introduced respectively, namely the expectation assumption and the dominating-term assumption.

Explicit universal sampling sets in finite vector spaces

In this paper we construct explicit sampling sets and present reconstruction algorithms for Fourier signals on finite vector spaces G , with | G | = p r for a suitable prime p . The two sampling sets have sizes of order O ( p t 2 r 2 ) and O ( p t 2 r 3 log ⁡ ( p ) ) respectively, where t is the number of large coefficients in the Fourier transform. The algorithms approximate the function up to a small constant of the best possible approximation with t non-zero Fourier coefficients. The fastest of the algorithms has complexity O ( p 2 t 2 r 3 log ⁡ ( p ) ) .

Sparse sums of squares on finite abelian groups and improved semidefinite lifts

Let G be a finite abelian group. This paper is concerned with nonnegative functions on G that are sparse with respect to the Fourier basis. We establish combinatorial conditions on subsets S and T of Fourier basis elements under which nonnegative functions with Fourier support S are sums of squares of functions with Fourier support T. Our combinatorial condition involves constructing a chordal cover of a graph related to G and S (the Cayley graph Cay($\hat{G}$,S)) with maximal cliques related to T. Our result relies on two main ingredients: the decomposition of sparse positive semidefinite matrices with a chordal sparsity pattern, as well as a simple but key observation exploiting the structure of the Fourier basis elements of G. We apply our general result to two examples. First, in the case where $G = \mathbb{Z}_2^n$, by constructing a particular chordal cover of the half-cube graph, we prove that any nonnegative quadratic form in n binary variables is a sum of squares of functions of degree at most $\lceil n/2 \rceil$, establishing a conjecture of Laurent. Second, we consider nonnegative functions of degree d on $\mathbb{Z}_N$ (when d divides N). By constructing a particular chordal cover of the d'th power of the N-cycle, we prove that any such function is a sum of squares of functions with at most $3d\log(N/d)$ nonzero Fourier coefficients. Dually this shows that a certain cyclic polytope in $\mathbb{R}^{2d}$ with N vertices can be expressed as a projection of a section of the cone of psd matrices of size $3d\log(N/d)$. Putting $N=d^2$ gives a family of polytopes $P_d \subset \mathbb{R}^{2d}$ with LP extension complexity $\text{xc}_{LP}(P_d) = \Omega(d^2)$ and SDP extension complexity $\text{xc}_{PSD}(P_d) = O(d\log(d))$. To the best of our knowledge, this is the first explicit family of polytopes in increasing dimensions where $\text{xc}_{PSD}(P_d) = o(\text{xc}_{LP}(P_d))$.

A note on small gaps between nonzero Fourier coefficients of cusp forms

It is shown that there are infinitely many primitive cusp forms f of weight 2 with the property that for all X large enough, every interval (X, X + cX(1/4)), where c > 0 depends only on the form, contains an integer n such that the n-th Fourier coefficient of f is nonzero.