Given a non-negative sequence $p=(p_i)_{i\in\mathbb{N}}\in \ell_1$ such that $\lVert p\rVert_1 = 1$,we define the two following quantities, for every $\varepsilon \in (0,1]$.

- Assuming, without loss of generality, that $p$ is non-increasing, let $k \geq 1$ be the smallest integer such that $\sum_{i\geq k} \leq \varepsilon$. Then we define $$ \Phi(\varepsilon, p) := \left( \sum_{i=2}^{k-1} p_i^{2/3} \right)^{3/2} \tag{1} $$ i.e., the $2/3$-quasinorm $\lVert p_{-\varepsilon}^{-\max{}} \rVert_{2/3}$ of the vector $p^{-\max{}}_{-\varepsilon}$ obtained by removing the largest element and the $\varepsilon$-tail of $p$.
- Defining, for $t>0$, the $K$-functional between $\ell_1$ and $\ell_2$ $$ \kappa_p(t) = \inf_{a+b=p} \lVert a\rVert_1 + t \rVert b\rVert_2 $$ and letting $$ \Psi(\varepsilon, p) := \kappa_p^{-1}(1-\varepsilon) \tag{2} $$ (right inverse, IIRC)

then can we prove upper and lower bounds relating (1) and (2)? Recent works of Valiant and Valiant [1] and Blais, Canonne, and Gur [2] imply such relation **[see at the end]** in a rather roundabout way (for $p$'s nontrivially point masses, i.e., say, $\lVert p\rVert_2 < 1/2$) *(by showing both quantities "roughly characterize" the sample complexity of a particular hypothesis testing problem on $p$ seen as a discrete probability distribution)*, but a **direct** proof of such a relation isn't known (at least to me), even only a loose one.

Is there a direct proof relating (upper and lower bounds) $\Phi(\cdot, p)$ and $\Psi(\cdot, p)$, of the form $$ \forall p \text{ s.t. } \lVert p\rVert_2 \ll 1,\forall x, \qquad x^\alpha \Phi(c x, p) \leq \Psi(c x, p) \leq x^\beta \Phi(C x, p) $$ ?

**A third related quantity, the $Q$-norm:** [2] (following some previous work of Montgomery-Smith [3]) does show a relation between (2) and a third quantity interpolating between $\ell_1$ and $\ell_2$ norms,
$$
T\in\mathbb{N} \mapsto \lVert p\rVert_{Q(T)} := \sup\{ \sum_{j=1}^T \left( \sum_{i\in A_j p_i^2 }\right)^{1/2} A_1,\dots,A_T \text{ partition of }\mathbb{N}\} \tag{3}
$$
as, for all $t>0$ such that $t^2\in \mathbb{N}$,
$$
\lVert p\rVert_{Q(t^2)} \leq \kappa_p(t) \leq \lVert p\rVert_{Q(2t^2)}.
$$

**The inequalities established in a roundabout way:**
The relation between the two quantities shown by combining [1] and [2] (which both establish sample complexity upper and lower bounds for the same hypothesis testing problem) can be rewritten as
$$
\forall p, \forall \varepsilon \in(0,1/9),\qquad 1\vee \Phi(9\varepsilon,p) \lesssim 1\vee \Psi(\varepsilon,p)
\lesssim 1\vee \frac{\Phi(\varepsilon/32,p)}{\varepsilon} \tag{4}
$$
where $\lesssim$ is an inequality, up to some (absolute) positive constant factor.

[1] Gregory Valiant and Paul Valiant. *An Automatic Inequality Prover and Instance Optimal Identity Testing.* SIAM Journal on Computing 46:1, 429-455. 2017.

[2] Eric Blais, Clément Canonne, and Tom Gur. *Distribution testing lower bounds via reductions from communication complexity.*
ACM Transactions on Computation Theory (TOCT), 11(2), 2019.

[3] Stephen J. Montgomery-Smith. *The distribution of Rademacher sums.* Proceedings of the American Mathematical Society, 109(2):517–522, 1990.