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 following two quantities, for each $ varepsilon in (0,1)

$.

- Assuming, without loss of generality, that $ p $ It's not growing, come on $ 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}

$$

that is, the $ 2/3 $-quasinorm $ lVert p _ {- varepsilon} ^ {- max {}} rVert_ {2/3} $ 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 $ Y $ ell_2 $

$$

kappa_p (t) = inf_ {a + b = p} lVert a rVert_1 + t rVert b rVert_2

$$

and leaving

$$

Psi ( varepsilon, p): = kappa_p ^ {- 1} (1- varepsilon) tag {2}

$$

(right reverse, IIRC)

So, can we prove the upper and lower limits by relating (1) and (2)? Recent works of Valiant and Valiant [1] and Blais, Canonne, and Gur [2] imply this relationship in a rather indirect way (for $ p $The non-trivial point masses, that is, say, $ lVert p rVert_2 <1/2 $) *(By showing that both quantities "approximately characterize" the complexity of the sample of a particular hypothesis test problem in $ p $ seen as a discrete probability distribution)*, but a **direct** the proof of such a relationship is not known (at least for me), even if it is only a loose one.

Is there a related direct test (upper and lower limits)? $ Phi ( cdot, p) $ Y $ Psi ( cdot, p) $, from the way

$$

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)

$$

?

[2] (Following some previous work by Montgomery-Smith [3]) shows a relationship between (2) and a third quantity that is interpolated between $ ell_1 $ Y $ ell_2 $ rules

$$

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} 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)}.

$$

[1] Gregory Valiant and Paul Valiant. *An automatic inequality tester and an optimal identity test instance.* SIAM Journal on Computing 46: 1, 429-455. 2017

[2] Eric Blais, ClĂ©ment Canonne and Tom Gur. *Distribution of lower limit tests by reducing the complexity of the communication.*

ACM Transactions on Computation Theory (TOCT), 11 (2), 2019.

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