Converse to Hausdorff-Young (or Riesz-Thorin) for finite cyclic groups?

Let $$v$$ be a vector $$v \in \mathbb{R}^p$$, with non-negative entries and $$p$$ prime. The Hausdorff-Young inequality gives bounds of the form: $$\|\mathcal{F}v\|_a \le C_{a,b} \|v\|_b$$ where the constants $$C_{a,b}$$ are explicitly known. There is a nice paper that gives an explicit characterization of all the $$v$$ for which this inequality is achieved. The proof is to break up the Riesz square into three different "regimes"; understand the extreme points; and then use Riesz-Thorin to interpolate to the rest of the square.

My question: I would like a quantitative understanding of what $$v$$ looks like when this inequality is far from sharp. Is there a quantitative version of the Hausdorff-Young inequality or Reisz interpolation theorem?

It appears Michael Christ showed exactly this, for functions over $$\mathbb{R}^d$$ instead of over $$\mathbb{Z}_p$$, https://arxiv.org/pdf/1406.1210.pdf. I am looking for an analogue for finite cyclic groups. (Not sure if this is easier or harder).

• Thank you for the catches! Edited. Regarding your first question; yes my group structure is just the finite cyclic group. – DJA Jan 27 at 19:43