arXiv subjects
Publications and source records attributed to Felipe Gonçalves.
Motivated by the recent paper [Sharp endpoint extension inequalities for the moment curve on finite fields II: an extremal property of the uniform distribution, arXiv:2609.29882], which proves sharp extension inequalities in finite fields via a two-point symmetrization argument, we prove here a more general component-wise inequality for permutation matches that implies theirs.
We investigate the maximization problem for the family of mixed-norm Strichartz inequalities for the Schrödinger equation, $\|e^{-itΔ/2}f\|_{L_t^qL_{\boldsymbol{x}}^r(\mathbb{R}^{1+d})}\le C_{q,r}\lVert f\rVert_{L^2(\mathbb{R}^d)}$, with $2/q+d/r=d/2$, $q,r\geq 2$, and thus $r\leq 2d/(d-2)$ if $d\geq 3$. We show that, in low dimensions $1\leq d\leq 5$, the thresholds $ρ_1=10$, $ρ_2=6$, $ρ_3=4\sqrt{7}-6\approx 4.583$, $ρ_4=2\sqrt{15}-4\approx 3.746$, and $ρ_5=10/3\approx 3.333$ are such that gaussians are stable local maximizers for $2<r<ρ_d$, and fail to be local maximizers for $ρ_d<r\leq 2d/(d-2)$ (with the conventions there is no upper bound on $r$ when $d\in\{1,2\}$ and that $r=\infty$ is excluded when $d=2$). In the cases $(q,r,d)\in\{(6,6,1),(8,4,1),(4,4,2)\}$, we establish global stability inequalities with effective stability constants. Both proofs hinge on spectral gaps which we compute exactly.
We study the problem of finding the norm of the point evaluation operator in the Paley--Wiener space $PW^{1}(\r^d)$, consisting of $d$-variable functions of spherical exponential type that are integrable on $\r^d$. The extremal functions can be taken radial, which naturally leads us to consider a related extremal problem in a weighted Paley--Wiener space of single-variable functions. We establish that the radial extremal function must satisfy a third-order linear ODE with polynomial coefficients for every $d \geq 1$, extending Gorbachev's recent odd-dimensional result. Along the way, we prove interpolation and reciprocal formulas involving the zeros of the extremizer.
We establish a convex order comparison for products of averages of exchangeable nonnegative random variables. As a consequence, we solve a conjecture of Lamkin and Tkocz (2022) on the log-convexity of moments of sample means, as well as its pointwise version, in a more general form involving arbitrary integer partitions.
We prove a sharp lower bound for the cardinality of sumsets of subsets of $\mathbb{Z}^d$ confined to a hypercube, resolving in strong form a conjecture that was made explicit by Becker, Ivanisvili, Krachun and Madrid and had circulated in the folklore of the field for some time. Specifically, for sets $A_j\subseteq \{0,1,2,\dots,m\}^d$ we show that \[|A_1+\dots+A_n|\;\geq\; (|A_1|\cdots|A_n|)^{1/p},\qquad p=\frac{n\log(m+1)}{\log(nm+1)},\] with the exponent best possible. The only previously known sharp cases were $A_j\subseteq \{0,1\}^d$, for all $n\ge1$, and $A_j\subseteq \{0,1,2\}^d$ for $n=2$. We also prove a sharp inequality in the case when $A_j\subseteq\{0,1,\dots,m_j\}^d$ for different $m_j$. We obtain the above inequality as a corollary of a stronger result on sup-convolution of functions on $\mathbb{Z}^d$, whose proof is based on a novel mixed volume representation of a lattice path norm, together with a sharp one-dimensional functional inequality.
We show that the extension map \[ \mathcal{E}_{NS}(f)(z)=\frac{f(x+y)+f(x-y)}{2}+i\frac{f(x+y)-f(x-y)}{2}\mbox{ for all }z=x+iy\in\mathbb{H}\,, \] defined by Norton and Sullivan in '96, is the only locally linear extension map taking bi-Lipschitz functions on $\mathbb{R}$ to quasiconformal functions on $\mathbb{H}$, modulo the action of a group isomorphic to the linear group. In fact, we discovered many other extension like this one (lying in the orbit of such group action), such as: $f(x)\mapsto f(x)+i(f(x)-f(x-y))$.
We completely classify Fourier summation formulas, and in particular, all crystalline measures with quadratic decay. Our classification employs techniques from almost periodic functions, Hermite-Biehler functions, de Branges spaces and Poisson representation. We show how our classification generalizes recent results of Kurasov \& Sarnak and Olevskii \& Ulanovskii. As an application, we give a new classification result for nonnegative measures with uniformly discrete support that are bounded away from zero on their support. Moreover, we give a new construction using eta-quotients, generalizing an old example of Guinand.
We prove a sharp Fourier extension inequality on the circle for the Tomas-Stein exponent for functions whose spectrum $\{\pm λ_n\}$ satisfies $λ_{n+1}>3 λ_{n}$.
We completely classify Fourier summation formulas of the form $$ \int_{\mathbb{R}} \widehatφ(t) dμ(t)=\sum_{n=0}^{\infty} a(λ_n)φ(λ_n), $$ that hold for any test function $φ$, where $\widehatφ$ is the Fourier transform of $φ$, $μ$ is a fixed complex measure on $\mathbb{R}$ and $a:\{λ_n\}_{n\geq 0}\to\mathbb{C}$ is a fixed function. We only assume the decay condition $$ \int_{\mathbb{R}} \frac{d |μ|(t)}{(1+t^2)^{c_1}} + \sum_{n\geq 0} |a(λ_n)|e^{-c_2 |λ_n|}<\infty, $$ for some $c_1,c_2>0$. This completes the work initiated by the first author previously, where the condition $c_1\leq 1$ was required. We prove that any such pair $(μ,a)$ can be uniquely associated with a holomorphic map $F(z)$ in the upper-half space that is both almost periodic and belongs to a certain higher index Nevanlinna class. The converse is also true: For any such function $F$ it is possible to generate a Fourier summation pair $(μ,a)$. We provide important examples of such summation formulas not contemplated by the previous results, such as Selberg's trace formula.
We give universal bounds on the fraction of nontrivial zeros having given multiplicity for L-functions attached to a cuspidal automorphic representation of $\mathrm{GL}_m/\mathbb{Q}$. For this, we apply the higher-level correlation asymptotic of Hejhal and Rudnick & Sarnak in conjunction with semidefinite programming bounds.
We study the sphere packing problem in Euclidean space where we impose additional constraints on the separations of the center points. We prove that any sphere packing in dimension $48$, with spheres of radii $r$, such that no two centers $x_1$ and $x_2$ satisfy $\sqrt{\tfrac{4}{3}} < \frac{1}{2r}|x_1-x_2| <\sqrt{\tfrac{5}{3}}$, has center density less or equal than $(3/2)^{24}$. Equality occurs for periodic packings if and only if the packing is given by a $48$-dimensional even unimodular extremal lattice. This shows that any of the lattices $P_{48p},P_{48q},P_{48m}$ and $P_{48n}$ are optimal for this constrained packing problem, and gives evidence towards the conjecture that extremal lattices are optimal unconstrained sphere packings in $48$ dimensions. We also provide results for packings up to dimension $d\leq 1200$, where we impose constraints on the distance between centers and on the minimal norm of the spectrum, showing that even unimodular extremal lattices are again uniquely optimal. Moreover, in the one-dimensional case, where it is not at all clear that periodic packings are among those with largest density, we nevertheless give a condition on the set of constraints that allows this to happen, and we develop an algorithm to find these periodic configurations by relating the problem to a question about dominos.
We prove an asymptotically sharp version of the Bourgain-Clozel-Kahane and Cohn-Gonçalves sign uncertainty principles for polynomials of sublinear degree times a Gaussian, as the dimension tends to infinity. In particular, we show that polynomials whose degree is sublinear in the dimension cannot improve asymptotically on those of degree at most three. This question arises naturally in the study of both linear programming bounds for sphere packing and the spinless modular bootstrap bound for free bosons.
We establish a sharp adjoint Fourier restriction inequality for the end-point Tomas-Stein restriction theorem on the circle under a certain arithmetic constraint on the support set of the Fourier coefficients of the given function. Such arithmetic constraint is a generalization of a $B_3$-set.
We prove new sign uncertainty principles which vastly generalize the recent developments of Bourgain, Clozel & Kahane and Cohn & Gonçalves, and apply our results to a variety of spaces and operators. In particular, we establish new sign uncertainty principles for Fourier and Dini series, the Hilbert transform, the discrete Fourier and Hankel transforms, spherical harmonics, and Jacobi polynomials, among others. We present numerical evidence highlighting the relationship between the discrete and continuous sign uncertainty principles for the Fourier and Hankel transforms, which in turn are connected with the sphere packing problem via linear programming. Finally, we explore some connections between the sign uncertainty principle on the sphere and spherical designs.
If dividing by $p$ is a mistake, multiply by $q$ and translate, and so you'll live to iterate. We show that if we define a Collatz-like map in this form then, under suitable conditions on $p$ and $q$, almost all orbits of this map attain almost bounded values. This generalizes a recent breakthrough result of Tao for the original Collatz map (i.e., $p=2$ and $q=3$). In other words, given an arbitrary growth function $N\mapsto f(N)$ we show that almost every orbit of such map with input $N$ eventually attains a value smaller than $f(N)$.
We discuss on Heisenberg uniqueness pairs for the parabola given by discrete sequences along straight lines. Our method consists in linking the problem at hand with recent uniqueness results for the Fourier transform.
In this paper we investigate a high dimensional version of Selberg's minorant problem for the indicator function of an interval. In particular, we study the corresponding problem of minorizing the indicator function of the box $Q_{N}=[-1,1]^N$ by a function whose Fourier transform is supported in the same box $Q_N$. We show that when the dimension is sufficiently large there are no minorants with positive mass and we give an explicit lower bound for such dimension. On the other hand, we explicitly construct minorants for dimensions $1,2,3,4$ and $5$ and, as an application, we use them to produce an improved diophantine inequality for exponential sums.
In this note we study the eigenvalue problem for a quadratic form associated with Strichartz estimates for the Schrödinger equation, proving in particular a sharp Strichartz inequality for the case of odd initial data. We also describe an alternative method that is applicable to a wider class of matrix problems.