• Corpus ID: 235658724

On classical inequalities for autocorrelations and autoconvolutions

@inproceedings{Pont2021OnCI,
  title={On classical inequalities for autocorrelations and autoconvolutions},
  author={Jaume de Dios Pont and Jos'e Madrid},
  year={2021}
}
In this paper we study an autocorrelation inequality proposed by Barnard and Steinerberger [1]. The study of these problems is motivated by a classical problem in additive combinatorics. We establish the existence of extremizers to this inequality, for a general class of weights, including Gaussian functions (as studied by the second author and Ramos) and characteristic function (as originally studied by Barnard and Steinerberger). Moreover, via a discretization argument and numerical analysis… 
View PDF on arXiv

Figures and Tables from this paper

References

SHOWING 1-10 OF 24 REFERENCES
Improved bounds on the supremum of autoconvolutions
The supremum of autoconvolutions, with applications to additive number theory
We adapt a number-theoretic technique of Yu to prove a purely analytic theorem: if f(x) is in L^1 and L^2, is nonnegative, and is supported on an interval of length I, then the supremum of the…
Decoupling for fractal subsets of the parabola
Abstract. We consider decoupling for a fractal subset of the parabola. We reduce studying lL decoupling for a fractal subset on the parabola tpt, tq : 0 ď t ď 1u to studying l2Lp{3 decoupling for the…
Hermite polynomials, linear flows on the torus, and an uncertainty principle for roots
EXTENSIONS OF AUTOCORRELATION INEQUALITIES WITH APPLICATIONS TO ADDITIVE COMBINATORICS
In a 2019 paper, Barnard and Steinerberger show that for $f\in L^1(\mathbf{R})$, the following autocorrelation inequality holds: \begin{equation*} \min_{0 \leq t \leq 1} \int_\mathbf{R} f(x) f(x+t)\…
On Suprema of Autoconvolutions with an Application to Sidon sets
TLDR
This work derives a relaxation of the problem that reduces to a finite number of cases and yields slightly stronger results, and should be able to prove lower bounds that are arbitrary close to the sharp result.
The Symmetric Subset Problem in Continuous Ramsey Theory
TLDR
This paper establishes upper and lower bounds for Δ(ε) of the same order of magnitude, and implies that every B*[g] set contained in {1, 2, . . . , n} has cardinality less than 1.
Three convolution inequalities on the real line with connections to additive combinatorics
On the Convolution Inequality f ≥ f ⋆ f
We consider the inequality f > f ? f for real functions in L1(Rd) where f ? f denotes the convolution of f with itself. We show that all such functions f are non-negative, which is not the case for…
A course in abstract harmonic analysis
Banach Algebras and Spectral Theory Banach Algebras: Basic Concepts Gelfand Theory Nonunital Banach Algebras The Spectral Theorem Spectral Theory of *-Representations Von Neumann Algebras Notes and…
...
1
2
3
...