Extensions of Autocorrelation Inequalities With Applications to Additive Combinatorics

Abstract

In a 2019 paper, Barnard and Steinerberger show that for f ∈ L 1 (R), the following autocorrelation inequality holds:L 1 , where the constant 0.411 cannot be replaced by 0.37. In addition to being interesting and important in their own right, inequalities such as these have applications in additive combinatorics where some problems, such as those of minimal difference basis, can be encapsulated by a convolution inequality similar to the above integral. Barnard and Steinerberger suggest that future research may focus on the existence of functions extremizing the above inequality (which is itself related to Brascamp-Lieb type inequalities).We show that for f to be extremal under the above, we must have Date: January 9, 2020. 2010 Mathematics Subject Classification. 26D10 (primary), 39B62 (secondary).

On optimal autocorrelation inequalities on the real line

“…It has recently come to our knowledge that, in the recent manuscript [8], the authors investigate properties that an extremizer to (1.6) must necessarily fulfill. Although they do not prove existence of extremizers, we believe that a suitable combination of their methods with ours may result in further progress towards lowering the constant in Theorem 1.5 towards the best constant.…”

Generalized difference sets and autocorrelation integrals

“…The continuous analog has received more attention, and we know that [1]). See also the discussion in [8].…”