# Gauss Sums and Hausdorff-Young

GaussSums-Hausdorff Young

Let $1 \leq p \leq 2$. The classical Hausdorff-Young inequality asserts that for any $f \in L^p(\mathbb{T})$, there is a $A_p$ (equal to 1 in the present case) such that $||\hat{f}||_{p/(p-1)} \leq A_p||f||_p$. There are examples that show one cannot allow $p > 2$. In the above pdf, we provide a construction, motivated from number theory, that shows $p > 2$ is impossible in the Hausdorff-Young inequality. One can think of this as a pseudo-random approach, as opposed to the random approach employed by, for instance, an application of Khintchine’s inequality.

We will first prove a discrete analog. For those a bit rusty, you are invited to check out my notes on the discrete Fourier transform.

I’d like to thank to of my fellow graduate students, Derek Jung and Xiao Li, for pointing out some mistakes in a previous version. I would also like to thank Sergei Konyagin for making me aware of the Shapiro-Rudin polynomials.

Lastly, I have not seen this example in the literature. If you have seen it, please let me know.