Sharp lower constants in the Khintchine inequality

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Numbers

$p$

$A_p$

1/6:

0.02209708691207961013752638631577653247765112305276481364338562090610519497597042248203730522386939957

1/5:

0.04419417382415922027505277263155306495530224610552962728677124181221038995194084496407461044773879915

1/4:

0.08838834764831844055010554526310612991060449221105925457354248362442077990388168992814922089547759830

1/3:

0.1767766952966368811002110905262122598212089844221185091470849672488415598077633798562984417909551966

2/5:

1/4

comment: The exponent $1/2-1/p$ is an integer, so this value is exact.

1/2:

0.3535533905932737622004221810524245196424179688442370182941699344976831196155267597125968835819103932

3/5:

0.4454493590701696523701131027952562539936063079390802016918784961259654016578812463187628836252785988

2/3:

1/2

comment: The exponent $1/2-1/p$ is an integer, so this value is exact.

3/4:

0.5612310241546864907167665248395897581162055553069933767202047729414520027829306235439616135562545404

4/5:

0.5946035575013605333587499852802379576464860462319087065095011123597333341134585799353906722690688369

5/6:

0.6155722066724581422496965345838715549380688805504088971685327691230503598596792292020113748254470708

0.7071067811865475244008443621048490392848359376884740365883398689953662392310535194251937671638207864

comment: $A_1=1/\sqrt{2}$, the sharp first-moment constant.

7/6:

0.7807091821557100915662087529289000436214944705186434469579333067692475167773281278851758207261616409

6/5:

0.7937005259840997373758528196361541301957466639499265049041428809126082528121095866367721066311104785

5/4:

0.8122523963562355226097093827752816512852049744321594425324937130542207434350148856247361298160288275

4/3:

0.8408964152537145430311254762332148950400342623567845108132260859749247549539022398143240041992925362

7/5:

0.8619728212469777242192933639474666076981029217257664217377692602612058740832835515905859187530500644

3/2:

0.8908987181403393047402262055905125079872126158781604033837569922519308033157624926375257672505571975

8/5:

0.9170040432046712317435415947941444280386551664368397497916620693532388311223473628414654893806603137

5/3:

0.9330329915368074159813432661499421670272299643514940389004973854853970203105635476037074733946453136

12/7:

0.9438743126816934966419131566675343760075683033387428137421251423182864877387067030451984546106177082

7/4:

0.9516951530106196099298932213328400239268288247496538509197976907046331899089615307296167803847626432

16/9:

0.9576032806985736469363056351479154439251382065557010735455510589062596528896244369183554159129788244

9/5:

0.9622238368941451396373408016638896438921847641104203143996269976146306924533906478517162827927175140

20/11:

0.9659363289248455510651443129204638993907373128792525806668938605719411049010477379328897192088814122

11/6:

0.9689844739012624397729118463988302081651568351548343054109579610446185447934186065004533732272721303

24/13:

0.9715319411536058687432894158212517892246100222689675445485405243173435782241536123786191394289894835

13/7:

0.9736036977213536290124855437548547709550572365629593036884540797452144368839072375015532370583532738

28/15:

0.9753849730761358216262870956671865389436896028981105969254922095032707379148516845591317955895221777

15/8:

0.9769410103450688937163896279578018043250041544226659985951805480651303390606497187346625459995543241

32/17:

0.9783119947882982064803860990976214385153146866951214883934180829160320697276372329894025410700847835

17/9:

0.9795290879738267692823876927213407963780631791576844902332331787976922588027708342878692187324804997

36/19:

0.9806168262222609757629723157693266445976457011025458983438387911620582894392677191579212676449857872

19/10:

0.9815947926151019485784989874356835284058993370873439506032481637617784327406511338886988733044836879

40/21:

0.9824788069415334396371616836963881628196114283983013858797994757473824218714680381297921707099701618

comment: $A_2=1$ follows from $\mathbb E(\sum_j a_j\varepsilon_j)^2=\sum_j a_j^2$.

Definition

For $p>0$, the sharp lower Khintchine constant [3] is the largest $A_p$ such that $A_p(\sum_j a_j^2)^{1/2}\leq(\mathbb E|\sum_j a_j\varepsilon_j|^p)^{1/p}$ for every finite real coefficient vector $a$.

Parameters

$p$

— moment exponent ($p>0$)

Formulas

(1)

$A_p=\min\{1,2^{1/2-1/p},G(p)\}$, where $G(p)=\sqrt{2}\left(\Gamma((p+1)/2)/\sqrt{\pi}\right)^{1/p}$ and $\Gamma$ is the Gamma function. All powers are positive real powers. [1], Theorem 1.1.

(2)

$A_p=1$ for $p\geq2$.

(3)

$\mathbb E|\varepsilon_1+\varepsilon_2|^p=2^{p-1}$ and $G(p)^p=\sqrt{2/\pi}\int_0^\infty x^p e^{-x^2/2}\,dx$.

Comments

(4)

The $\varepsilon_j$ are independent signs with probabilities $1/2$, and expectation is a probability average. The constant is defined for every real $p>0$; this table records rational exponents. Vector length is universally quantified. The entries are the lower constants themselves, not their reciprocals or powers.

(5)

The two-point expression $2^{1/2-1/p}$ is the ratio of $(\mathbb E|\varepsilon_1+\varepsilon_2|^p)^{1/p}$ to $\sqrt{2}$. The Gaussian expression $G(p)$ equals $(\mathbb E|Z|^p)^{1/p}$ for a standard real normal variable $Z$ and is the limit for equal coefficients $a_j=1/\sqrt{n}$ as $n\to\infty$.

(6)

The exponents $p=2m/(m+1)$ occur in estimates for multilinear Bohnenblust-Hille constants, [2], Proposition 3.1. The constant denoted $A_{\mathbb R,p}$ in that source is $1/A_p$ in this table.

Programs

(P1)

Sage

import numberdb.sage as numberdb
from sage.rings.rational_field import QQ
from sage.rings.real_arb import RealBallField

def A(p, digits=100):
    p = QQ(p)
    if p <= 0:
        raise ValueError("p must be positive")
    if p >= 2:
        return QQ(1)
    R = RealBallField(numberdb.bits(digits, losing=96))
    e = QQ(1)/2 - QQ(1)/p
    t = (R(2).log()*R(e)).exp()
    g = (R(2).log()/2 + (R((p+1)/2).gamma().log() - R.pi().log()/2)/R(p)).exp()
    if t < g and t < 1:
        return QQ(2)**int(e) if e.denominator() == 1 else t
    if g < t and g < 1:
        return g
    raise ArithmeticError("Increase precision to separate the branches")

A(QQ(13)/7)

References

[1]

O. Mordhorst, The optimal constants in Khintchine's inequality for the case $2<p<3$, Colloquium Mathematicum 147 (2017), 203–216. (doi)

[2]

F. Bayart, D. Pellegrino and J. B. Seoane-Sepúlveda, The Bohr radius of the $n$-dimensional polydisk is equivalent to $\sqrt{(\log n)/n}$, Advances in Mathematics 264 (2014), 726–746. (arXiv) (doi)

Links

[3]

Mordhorst: sharp constants in Khintchine's inequality (full text)

Similar tables

Values of the Gamma function at rational numbers — provides the Gamma values entering the Gaussian branch of $A_p$

Data properties

Entries are of type: real number

digits: 100 (Unknown key)

How well the digits are known: proven

Table is complete: no (it holds $p=1,2$, the application exponents $p=2m/(m+1)$ for $2\leq m\leq20$, and every reduced fraction $0<a/b<2$ with $b\leq6$, with overlaps removed; the application cutoff covers early cases and both nonconstant branches, while the noncanonical fraction bound supplies compact, notation-independent exploratory coverage, especially for $p<1$; values for $p>2$ are identically 1 and are described by (2))

How they were obtained:

Exact rational inputs are converted to Arb balls at 429 bits, comprising the 100-digit target and a 96-bit guard. Strict ball comparisons select the minimum in (1); no branch is chosen from rounded digits. Integer powers of 2 and the unit branch are returned exactly.

Every inexact computed value has relative radius less than $10^{-125}$, and its entire ball lies inside the stored 100-significant-digit decimal interval. Independent 150-digit quadrature of the Gaussian moment and explicit four-sign averages reproduce every row. The quadrature is a numerical cross-check, not the source of the proven enclosures. Gaussian moments of orders 0, 2 and 4 and known sign averages serve as controls. The attached checks also test the power identity on every two-point row and the defining inequality on five finite coefficient vectors at every selected exponent.