Sharp upper constants in the Khintchine inequality

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Numbers

$p$

$B_p$

13/6:

1.029965605409935206387000651877852361147345579224645150741177173325817331684718696072223680483800276

11/5:

1.035857547340306097160634160617261724100007723340119779836390836027851414323266767073586484732696638

9/4:

1.044634782580381597221296295423597326267711856637046006743763015667958282463077951740324009472826077

7/3:

1.059105956430157429249888362427296055213542291296521438251979377772279640221766612669330067739176411

12/5:

1.070545491251161847352181266284181035523852997298383371871826072143271058746385862442327263157495151

5/2:

1.087484427895791883602767466189897564879228281408480498881678354928924451475886127340419413571605179

13/5:

1.104169287797457109081656643897210394122253726900542351677262858635041575655796711040878847992349495

8/3:

1.115156790863555360189478775414071537770311780808035739801785027566418145290540316688151978095883383

11/4:

1.128743832582723017589374802717413659050126310656071186679667292807201770417932905136436872593629685

14/5:

1.136819650608013131551207103069024963089005627850699127403627835229286111098677240413908383208342588

17/6:

1.142172417827655186944203859946146558909119803929466961535715134733133428059821061207582743788948335

1.168575254962465548670476011097685271060524048167907972383516287423415293888785465278714234283834940

19/6:

1.194404613823215974691141502663558013029209736003007604852060789190806196917753898963629427003657828

16/5:

1.199504840728212800580040594347753071584753519256709978867113337723710507661045387576294212654134765

13/4:

1.207115418910280716560080887755696741103906901332713331665322917979183552762064126625264793655627905

10/3:

1.219695821606148543116829708523649192145728393641000483874792375643830132292819297556847908709641391

17/5:

1.229668868856967409287215652803532426992122447174379804861041238583056293187876964655265016344907877

7/2:

1.244480759488327894863323400495897862860215288174347623319900032320955278314834968915613813435927522

18/5:

1.259120841408089105921136791624485750187536979876670072727156717436904461779882966398001410341363273

11/3:

1.268788311586449557002864601372414816632810965904888625929375289820130382388512613689361654195377600

15/4:

1.280771368762240418283651510004228498283724386695380201654223195439851964654995985469860913492140827

19/5:

1.287908364566674602185895152514614483444076315605267011914759405059985248047230170336486041486727361

23/6:

1.292644741923970672164012232922937085544162095563518650617852510930017814777886097147880033286115545

1.316074012952492460819218901796999055160068590205822176731922658595866795197302133050743150246601932

9/2:

1.384009387510219688034827297145567581107625648979004481102158342577011288357288874498912543721018054

1.448791901549305285253546598812810588213401039351967807295030580155436284774272812054274028124363387

11/2:

1.510820744666641209063341845971947762906055014661744600117187967379529202284760321868134343756129018

1.570417802475019735310529667002623349458377776658135337896289938400746585292605422213434690489066223

13/2:

1.627847033598495929519685945457695828763318127356752743392050251575368960450554507646474448251247048

1.683328037076984812307523017899448565986396083702453214341564245770365627825987118778554845988423891

15/2:

1.737045801167462120104930792672689199283198315432927301438295319551549238786400958508621625378480723

1.789157866970849351206931327261384308137391537547955776686507592927051213997103356902180956570711662

17/2:

1.839799696435835697582595865580701029752330834575340914859907604220879317926403289485732461617688491

1.889088764782642822783312378939914427784245443077198685721622155407939950297877500434083066737976081

19/2:

1.937127728581304356044666688107505062745467029982011859718100916002051963496716328022716265813500627

10:

1.984006912043239739162771692048802797089379252760328830500687403034970101513601889418256688023929346

11:

2.074597036863298539479717135158140137390790258775003848374533918907752462423916465761367543547230813

12:

2.161401131429970669254059075385888153125359564002942746993505110195641258919793747111302677781655355

13:

2.244856871513839083720894766174114906836782958779970181719270810321461149050673125462911230795371243

14:

2.325323697606085246958117086044968495079807708388903801397300050907151369924179961761313848930174195

15:

2.403101075673934605087768426409801959792317818063567016041648042891881491206134611395237479985965092

16:

2.478441623213817297702015934269418157052545332578559794495167285167825389795473431085863561664969174

17:

2.551560760749834492474967731363328715355977516290471525302049808734010368908236687816181811244184913

18:

2.622643950021257850432598782401545227518433299298930862475163238736379259391963179554810710804651846

19:

2.691852217310018995864611811051317763729083634909054615500983895191869286769801202399203272329954249

20:

2.759326433478562223354647687451004524362548167293974660636915767542721572344762461688123259475781891

21:

2.825190676374429183830126904601035596969527945871978896357281911198913507609204307758475199892127089

22:

2.889554905058612493541517634885693731610308951062073559203314562816108549298365142543806703854448088

23:

2.952517110471649963303792642785762003265892960142068727848272139950893303689119516189520589591483179

24:

3.014165062566142046176411774522065545636406273616816610402052337742561719634752927228362785138545129

25:

3.074577742722480931240562729630927394908996449327158478400642808180641514238673626970671954211698993

26:

3.133826528055539222117966775576565310701116948264060988115403420140986880169540242521024291096361459

27:

3.191976178180336253940657466150037037528502162512810103810412611367567899726282794179899993069362736

28:

3.249085663261735534165155317026275333300950869190120224227452940467901233041627872844649420159937894

29:

3.305208863467718130786731755674425326027286422635449563382409273574431874709745453779359547165024042

30:

3.360395163417110605952303930109068226250084455569598633294086501092164303370244392438824688356236110

31:

3.414689960263908727773639560137024037546112023977976235851892482493894395594320351582067256506167967

32:

3.468135100271812140169040833862319830960506153269289991614266680064195712132945833449535519455096415

Definition

For $p>0$, the sharp upper Khintchine constant [2] is the smallest $B_p$ such that $(\mathbb E|\sum_j a_j\varepsilon_j|^p)^{1/p}\leq B_p(\sum_j a_j^2)^{1/2}$ for every finite real coefficient vector $a$, where the $\varepsilon_j$ are independent signs with probabilities $1/2$.

Parameters

$p$

— moment exponent ($p>0$)

Formulas

(1)

$B_p=1$ for $0<p\leq2$.

(2)

$B_p=\sqrt{2}\left(\Gamma((p+1)/2)/\sqrt{\pi}\right)^{1/p}$ for $p>2$, where $\Gamma$ is the Gamma function and every power is the positive real power. [1], Theorem 1.1.

(3)

$B_p^p=\sqrt{2/\pi}\int_0^\infty x^p e^{-x^2/2}\,dx$ for $p\geq2$.

(4)

$B_{2m}^{2m}=(2m-1)!!$ for integers $m\geq1$, where $(2m-1)!!=\prod_{j=1}^{m}(2j-1)$ is the number of pairings of $2m$ labelled points.

(5)

$B_{2m+1}^{2m+1}=2^{m+1/2}m!/\sqrt{\pi}$ for integers $m\geq1$.

(6)

$B_{p+2}^{p+2}=(p+1)B_p^p$ for $p\geq2$.

Comments

(7)

Expectation is a probability average. The constant is defined for every real $p>0$; this table records rational exponents. Vector length is universally quantified. Entries are the norm constants themselves.

(8)

For $p>2$, $B_p=(\mathbb E|Z|^p)^{1/p}$ for a standard real normal variable $Z$. The $L^p$ norms of the normalized equal-coefficient sums $n^{-1/2}\sum_{j=1}^n\varepsilon_j$ tend to $B_p$, so the constant is sharp over all finite vector lengths. [1], Section 2.

(9)

For $0<p\leq2$, monotonicity of probability $L^p$ means and $\mathbb E|\sum_j a_j\varepsilon_j|^2=\sum_j a_j^2$ give $B_p=1$; equality holds for a single nonzero coefficient.

(10)

The sharp lower Khintchine constants give the opposite side of the same inequality.

Programs

(P1)

Sage

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

def B(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))
    moment = R((p+1)/QQ(2)).gamma()/R.pi().sqrt()
    return R(2).sqrt()*(moment.log()/R(p)).exp()

B(QQ(7)/QQ(3))

References

[1]

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

Links

[2]

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

Similar tables

Sharp lower constants in the Khintchine inequality — gives the largest lower norm constant for the same real Rademacher sums

Values of the Gamma function at rational numbers — provides the Gamma values in the Gaussian moment formula for the upper constants

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 $p=1,2$, all integer exponents $3\leq p\leq32$, every reduced noninteger fraction $2<a/b<4$ with $b\leq6$, and the half-integers $9/2$ through $19/2$; integer moments form the factorial and pairing families, while the noncanonical fraction bounds provide compact reference coverage near the branch transition and a single fractional slice at moderate orders; the remaining unit plateau $0<p\leq2$ is specified by (1))

How they were obtained:

Exact rational exponents select the unit branch before any approximation. Other values are enclosed by Arb at 429 bits, comprising the 100-digit target and a 96-bit guard. Every computed inexact value has relative radius less than $10^{-125}$, and its entire ball is contained in the written 100-significant-digit decimal interval. The unit values are exact integers.

Independent 160-digit quadrature of the Gaussian density reproduces every nonconstant row; the quadrature corroborates the computation and is not the source of the certified error bounds. Known Gaussian moments of orders 0, 2 and 4 and finite-sign averages serve as controls. Integer moment identities, the two-step recurrence wherever both arguments are stored, and the defining inequality for five finite coefficient vectors were checked before submission and on database readback. The attached generator and argument-selection note specify the enumeration.