Diophantine Approximation and applications in Interference Alignment

Alignment

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Adiceam, Faustin, Beresnevich, Victor orcid.org/0000-0002-1811-9697, Levesley, Jason et

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Alignment. Advances in Mathematics. pp. 231-279. ISSN 0001-8708

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Contents lists available atScienceDirect

Advances

in

Mathematics

www.elsevier.com/locate/aim

Diophantine

approximation

and

applications

in

interference

alignment

F. Adiceam,V. Beresnevich,J. Levesley, S. Velani∗_{, E. Zorin}

DepartmentofMathematics,UniversityofYork,York,YO105DD,UK

a r t i c l e i n f o a b s t r a c t

Articlehistory:

Received8June2015

Receivedinrevisedform26June 2016

Accepted4July2016 Availableonlinexxxx

CommunicatedbyKennethFalconer

DedicatedtoMauriceDodson

MSC:

primary11J83

secondary11J13,11K60,94A12

Keywords:

MetricDiophantineapproximation Khintchine–GroshevTheorem Non-degeneratemanifolds

Thispaper ismotivatedby recentapplicationsof Diophan-tineapproximationinelectronics,inparticular,intherapidly developingareaofInterference Alignment.Someremarkable advancesinthisareagivesubstantialcredittothe fundamen-talKhintchine–GroshevTheoremand,inparticular,toitsfar reachinggeneralisationforsubmanifoldsofaEuclideanspace. Witha view towards the aforementionedapplications, here weintroduce andprovequantitative explicitgeneralisations oftheKhintchine–GroshevTheoremfornon-degenerate sub-manifoldsofRn

.Theimportanceofsuchquantitative state-mentsisexplicitlydiscussedinJafar’smonograph[12,§4.7.1]. ©2016TheAuthor(s).PublishedbyElsevierInc.Thisisan openaccessarticleundertheCCBYlicense (http://creativecommons.org/licenses/by/4.0/).

Contents

1. Introduction . . . 232 2. Thetheoryforindependentvariables . . . 236

✩

FAresearchissupportedbyEPSRCProgrammeGrant:EP/J018260/1.VBandSVresearchissupported inpartbyEPSRCProgrammeGrant:EP/M021858/1.

* Correspondingauthor.

E-mailaddresses:[email protected](F. Adiceam), [email protected]

(V. Beresnevich),[email protected](J. Levesley),[email protected](S. Velani),

[email protected](E. Zorin).

http://dx.doi.org/10.1016/j.aim.2016.07.002

3. Diophantineapproximationonmanifolds . . . 243

4. PreliminariesforTheorem 3 . . . 244

5. AstrengtheningandproofofTheorem 3. . . 249

6. ProofofTheorem 5 . . . 252

7. Verifyingconditions(i)&(ii)ofTheorem 7. . . 254

8. ProofofProposition 2. . . 268

Acknowledgments . . . 275

Appendix A. Diophantineapproximationonmanifoldsandwirelesstechnology . . . 275

References . . . 279

1. Introduction

The present paper is motivated by a recent series of publications, including [11,12, 14–17,21–23], whichutilise thetheory ofmetricDiophantine approximation todevelop new approaches in interference alignment, a concept within the field of wireless com-munication networks.This newlinkisbothsurprisingand striking.Thekeyingredient from the number theoretic side is the fundamentalKhintchine–Groshev Theorem and its variations. In this paper we seek to address certain problems in Diophantine ap-proximationwhichcropup,orimpingeupon,theapplicationstointerferencealignment. TheresultsobtainedrepresentquantitativerefinementsoftheKhintchine–Groshev The-orem thatarerelevantto the applications mentionedabove.Indeed, thedesirabilityof such quantitative statements is explicitly eluded to in Jafar’s monograph [12, §4.7.1]. While the main content of the paper is purely number theoretic, in Appendix A we attempt to illustrate at abasiclevel themanner inwhich Diophantine Approximation plays a natural role in Interference Alignment. The appendix is very much intended for thereader whose background is not inelectronics butis neverthelessinterested in applications.

Although the main emphasis will be on the Khintchine–Groshev Theorem for sub-manifolds ofRn _{[2,5,7,8,13,20],webegin byconsideringtheclassicaltheoryforsystems}

oflinearformsofindependentvariables.Thisapproachhastwo benefits.Firstly,weare able to introduce the key ideas without too much technical machinery obscuring the picture. Secondly, therefinements of theclassical theory produceeffective results with much betterconstants.

InordertorecallKhintchine’stheoremwefirstdefinetheset_W(ψ) ofψ-well approx-imable numbers.Tothisend,denotebyR_{+ thesetofnon-negative realnumbers.Given}

arealpositivefunctionψ:R_+→R_{+ withψ(r)→0 asr→ ∞,letthen}

W(ψ) :={x∈R_{:|qx−p|< ψ(q) for i.m. (q, p)∈}N_×Z_},

Asimple‘volume’argumenttogetherwiththeBorel–CantelliLemmafromprobability theoryimpliesthat

|W(ψ)_|= 0 if

∞

q=1

ψ(q) <_∞,

where_|X_|standsfortheLebesguemeasureofX_⊂R_{.Theaboveconvergencestatement}

representstheeasierpartofthefollowingbeautifulresultduetoKhintchinewhichgives acriterionforthesizeoftheset_W(ψ) intermsofLebesguemeasure.Inwhatfollows,we saythatX _⊂R_isfullinR_andwrite|X|=Full_if|R_{\X|= 0;thatis,thecomplement}

ofX inR_{isofLebesgue measurezero.The followingisaslightlymoregeneralversion}

ofKhintchine,see[4].

TheoremA(Khintchine, 1924). Letψbe anapproximatingfunction. Then

|W(ψ)_| =

⎧ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎩

0 if

∞

q=1

ψ(q)<_∞,

Full _if

∞

q=1

ψ(q) =_∞ andψ is monotonic.

Thus, given any monotonic approximating function ψ, for almost all1 _{x ∈ R the}

inequality_|x₋p/q_|< ψ(q)/q holdsforinfinitelymanyrationalnumbersp/qifandonly ifthesum∞_q=1ψ(q) diverges.

TherearevariousgeneralisationsofKhintchine’stheoremtohigherdimensions—see

[3] for an overview. Here we shall consider the case of systems of linear forms which originates from a paper by Groshev in 1938. In what follows, m and n will denote positive integers and Mm,n will stand for the set of m×n matrices over R. Given a

functionΨ:Zn_→R_+,let

Wm,n(Ψ) :=_{X= (xi,j)∈Mm,n:Xa<Ψ(a) for i.m.a∈Zn\ {0}},

wherea= (a1,. . . ,an),

Xa:= max

1≤i≤mxi,1a1+. . .+xi,nan

andx:= min_{|x₋k_|:k_∈Z_{}isthedistanceofx∈}R_{fromthenearestinteger.Given}

asubsetX inMm,n,wewillwrite|X|mnforitsambient(i.e.mn-dimensional)Lebesgue

measure.Itiseasilyseenthat_W1,1(Ψ) coincideswithW(ψ) whenΨ(q)=ψ(|q|).

There-fore the following result is the natural extension of Theorem A to higher dimensions. Noticethatthere isnomonotonicityassumptionontheapproximatingfunction.

Theorem B.Letm,n∈N_{with nm>1,ψ:}N_→R_{+ be anapproximatingfunctionand}

Σψ:=

∞

q=1

qn−1ψ(q)m. (1)

Let Ψ : Zn _→ R_{+ be given by Ψ(a) := ψ(|a|) for a = (a1,. . . ,an) ∈} Zn_{\ {0}, where} |a_|= max1≤i≤n|ai|.Then

|Wm,n(Ψ)_|mn =

⎧ ⎨ ⎩

0 if Σψ<∞,

Full _{if Σψ=∞.}

Theorem B was first obtained by Groshev under the assumption that qn_ψ(q)m _is

monotonic inthecaseof divergence.Theredundancy ofthemonotonicityconditionfor

n ≥ 3 follows from Schmidt’s paper [18, Theorem 2] and for n = 1 from Gallagher’s paper [10]. Theorem Bas statedwaseventuallyprovedin[6]wheretheremaining case of n = 2 was addressed. The convergence case of Theorem B is a relatively simple application of the Borel–Cantelli Lemmaand it holds for arbitrary functions Ψ. Thus togetherwithTheorem A,wehavethefollowingextremelygeneralstatementinthecase of convergence.

Theorem C.Letm,n_∈N_andΨ:Zn_→R_{+ be anyfunctionsuch thatthesum}

ΣΨ:=

a_∈Zn_\{0_}

Ψ(a)m (2)

converges. Then

|Wm,n(Ψ)_|mn= 0.

AnimmediateconsequenceofTheorem Cisthefollowing statement.

Corollary 1.LetΨ beas in Theorem C.Then, for almostevery X_∈Mm,n there exists

a constantκ(X)>0suchthat

Xa > κ(X) Ψ(a) _∀ a_∈Zn_{\ {0}. (3)}

Bm,n(Ψ, κ) :=X∈Mm,n:Xa> κΨ(a) ∀ a∈Zn\ {0} . (4)

Then,foranyκandΨ,thesetB1,n(Ψ,κ) willnotcontain

[₋κΨ(a), κΨ(a)]_×Rn−1

witha = (1,0,. . . ,0).Thisset is ofpositiveprobability.In thelightof thisexample it becomeshighlydesirabletoaddressthefollowing problem:

Problem.InvestigatethedependencebetweenκandtheprobabilityofBm,n(Ψ,κ).

Asthefirststeptounderstandingthisproblemweobtainthefollowingstraightforward consequenceofTheorem C.

Theorem 1.Let m,n _∈ N _{and μ be a probability measure on Mm,n that is absolutely}

continuouswithrespecttoLebesguemeasureonMm,n.LetΨ:Zn →R+beanyfunction

suchthat (2)converges.Thenforany δ∈(0,1)thereisaconstantκ>0depending only onμ,Ψandδ suchthat

μ(_Bm,n(Ψ, κ))_≥1₋δ. (5)

Priorto givingaproofofthistheorem recallthatameasureμonMm,nisabsolutely

continuouswithrespecttoLebesguemeasureifthereexistsaLebesgueintegrablefunction

f : Mm,n →R+ such that for every Lebesguemeasurable subset A of Mm,n, one has

that

μ(A) =

A

f, (6)

where _Af isthe Lebesgueintegralof f over A. Thefunction f is oftenreferredto as thedistribution(ordensity) of μ.

Proof. Since μis absolutely continuouswith respect to Lebesguemeasure,Theorem C

impliesthatμ(_Wm,n(Ψ))= 0.Henceμ(Mm,n\ Wm,n(Ψ))=μ(Mm,n)= 1.Note that

κ>0

Bm,n(Ψ, κ) =Mm,n\ Wm,n(Ψ).

Theorem 1nowfollowsonusingthecontinuityofmeasures. 2

submanifolds ofRn_{. This constitutes themain substance ofthe paper. Theresults are}

obtainedbyexploitingthetechniques ofBernik,Kleinbock andMargulis[8]originating from the seminalwork ofKleinbock and Margulis[13] onthe Baker–Sprindžuk conjec-ture.

2. Thetheoryforindependentvariables

TobeginwithwegiveanalternativeproofofTheorem 1whichintroducesanexplicit construction that will be utilised for quantifying thedependence of κon δ. Indeed, in thecasethatμisauniformdistributiononaunitcubetheproof alreadyidentifiesthe requireddependence.

2.1. Theorem 1revisited

Byaunitcube inMm,nwewillmeanasubsetofMm,ngivenby

(xi,j)∈Mm,n:αi,j≤xi,j< αi,j+ 1

for somefixed matrix(αi,j)∈Mm,n.Given a∈Zn\ {0}andε>0,letW(a,ε) denote

theset ofX_∈Mm,nsuchthat

Xa _≤ε . (7)

Itiseasilyseenthat_W(a,ε) isinvariantunderadditivetranslationsbyanintegermatrix; thatis,

W(a, ε) +B=_W(a, ε)

for any B _∈Mm,n(Z), where Mm,n(Z) denotesthe set of m×n matrices with integer

entries. Furthermore,wehavethat

|W(a, ε)∩P|mn= (2ε)m (8)

for any 0_≤ε _≤ 1

2 and any unit cube P inMm,n. This follows, for example, from [19,

Chapter 1, Lemma 8].Then,since

ΣΨ:=

a∈Zn_\{0}

Ψ(a)m_{<∞ (9)}

we musthavethat

Inwhatfollowswewill assumethat

2κMΨ≤1. (11)

Thisconditionensuresthatwecanapply(8)withε=κΨ(a). FixaunitcubeP0inMm,n andforeachΔ∈Mm,n(Z),let

P∆:=P0+ Δ

denotetheadditivetranslationofP0byΔ.Clearly,P∆itselfisaunitcube.Furthermore,

Mm,n=

◦

∆∈Mm,n(Z₎

P∆. (12)

Notethattheunionisdisjoint.Using(8)andthefactthat

Mm,n\ Bm,n(Ψ, κ) =

a_∈Zn_\{0}

W(a, κΨ(a)),

weobtainthatforeachΔ_∈Mm,n(Z),

|P∆\ Bm,n(Ψ, κ)|mn≤

a_∈Zn_\{0}

|W(a, κΨ(a))∩P∆|mn

=

a∈Zn_\{0}

(2κΨ(a))m = (2κ)mΣΨ. (13)

Since μ is aprobability measure, it follows from (12)that there exists a finite subset

A_⊂Mm,n(Z) suchthat

μ

∆∈A P∆

>1₋δ/2. (14)

Let N = #A be the number of elements in A. Since μ is absolutely continuous with respecttoLebesguemeasure,foreveryΔ∈Aandany ε1>0,thereexists ε2suchthat

foranymeasurable subsetX ofP∆,

|X_|mn< ε2 ⇒ μ(X)< ε1. (15)

In view of (13), applying (15) to X = P∆\ Bm,n(Ψ,κ) and ε1 = δ/(2N) implies the

existenceof

suchthat

μ(P∆\ Bm,n(Ψ, κ))< δ/(2N) if (2κ)mΣΨ≤ε2(Δ, δ, N). (16)

Inparticular,thesecondinequalityin(16)holdsif

κ≤κ∆:=1

2

ε2(Δ, δ, N)

ΣΨ

1/m .

Since Aisfinite,there existsκsatisfying(11)and

0< κ_≤min

∆∈Aκ∆.

Clearly,forsuchachoiceofκthefirstinequalityin(16)holdsforanyΔ∈A.Hence,by

(14)andtheadditivityofμweobtainthat

μ(Mm,n\ Bm,n(Ψ, κ))≤ δ

2+

∆∈A

μ(P∆\ Bm,n(Ψ, κ))

≤ δ₂+

∆∈A δ

2N = δ

2+N

δ

2N = δ .

Theupshot ofthisisthat

μ(_Bm,n(Ψ, κ)) = 1₋μ(Mm,n\ Bm,n(Ψ, κ))≥1−δ , (17)

whichcompletes theproofofTheorem 1.

2.2. Quantifyingthedependenceof κonδ

Wenowturnourattentiontoquantifyingthedependenceofκonδwithinthecontext ofTheorem 1.Tothisend,wewillmakeuseoftheLpnorm.GivenaLebesguemeasurable

functionf :Mm,n→R+,ameasurablesubsetXofMm,nandp≥1,wewritef ∈Lp(X)

iftheLebesgueintegral

X

|f_|p:=

Mm,n |f_|pχX

exists andis finite.Here χX isthe characteristicfunctionofX.Forf ∈Lp(X),theLp

norm off onX isdefinedby

f_p,X :=

⎛ ⎝

X |f_|p

⎞ ⎠

1/p

Inthecasep=∞,theL∞-normonX isdefinedastheessentialmaximumof|f|onX;

thatis,

f∞,X := inf{c∈R:|f(x)| ≤cfor almost allx∈X}.

Iff_∞,X <∞,then wewrite f ∈L∞(X).Forexample,iff iscontinuousand X isa

non-emptyopensubsetof Mm,n, thenf∞,X issimplythesupremumoff onX.The

followinglemma gatherstogethertwo wellknowfactsregardingtheLpnorm.

Lemma1.

(1) Forany p_≥1andanymeasurable subsets X_⊂Y,

f_{p,X ≤} f_p,Y.

(2) (Hölder’s inequality)Forany 1≤p,q≤ ∞ satisfying 1_p+1_q = 1,

X f g

≤ fp,Xgq,X.

Thenextlemma isacorollaryof Lemma 1.

Lemma 2.Letp>1and μbe a probabilitymeasure on Mm,n with density f.LetX be

aLebesguemeasurablesubset of Mm,n.If f ∈Lp(X),then

μ(X)_≤ f_p,X|X_|1_mn−1/p.

Proof. Bydefinition,we havethat

μ(X) =

X f .

Defineq bytheequation 1

p +1q = 1.ThenbyHölder’sinequality,wehavethat

μ(X) =

X

f_×1 _≤ f_p,X1_q,X = f_p,X ⎛ ⎝

X

1q

⎞ ⎠

1/q

≤ f_p,X|X_|1_mn−1/p

asrequired. 2

thatthereexists somep>1 suchthatforeveryΔ∈A,thedensityf ofμhasfiniteLp

norm onP∆.

Letκbesuchthat(11)issatisfied.Inthiscase,(13)holdsforeveryP∆withΔ∈A.By

Lemmas 1 and 2,

μ(P∆\ Bm,n(Ψ, κ))≤ fp,P∆· |P∆\ Bm,n(Ψ, κ)|

1−1/p

mn .

Using (13),we obtainthat

μ(P∆\ Bm,n(Ψ, κ))≤ fp,P∆·

(2κ)mΣΨ

1−1/p

(18)

where ΣΨ isgivenby(9).Itfollows that

μ(Mm,n\ Bm,n(Ψ, κ))≤ δ

2+

∆∈A

μ(P∆\ Bm,n(Ψ, κ))

≤ δ₂+(2κ)mΣΨ

1−1/p

Σf ≤ δ

if

κ≤1₂

Σ−_Ψ1

_δ

2Σf p

p−11/m ,

where

Σf :=

∆∈A

fp,P∆. (19)

SinceAisfinite,thequantityΣf isalsofinite.Theupshotoftheabovediscussionisthe

following statement.

Theorem 2(Effective version of Theorem 1).Let m,n_∈ N_{,μ, Ψ be as in Theorem 1,}

let MΨ be given by (10)and let f denote thedensity of μ.Furthermore, let P0 be any

unit cubein Mm,n andA beany finite subsetof Mm,n(Z)satisfying(14).Assumethere

exists p>1suchthat f ∈Lp(P∆)forany Δ∈Aand alsoassume thatthequantity Σf

is givenby(19).Then, foranyδ∈(0,1),inequality(5) holdswith

κ:= 1 2min

⎧ ⎨ ⎩

1

MΨ,

Σ−_Ψ1

_δ

2Σf p

p−11/m ⎫ ⎬

⎭. (20)

Remark1.InthecasewhenΨ iseven,thatisΨ(−a)= Ψ(a) foralla∈Zn_\{0},onecan

improveformula (20)forκbyreplacing ΣΨ with 1₂ΣΨ.This isanobvious consequence

of the fact that in this case the sets _W(a,κΨ(a)) and _W(₋a,κΨ(₋a)) coincide and thereforedonothavetobe countedtwicewithintheproof.

There are various simplifications and specialisations of Theorem 2 when we have extrainformationregardingthemeasureμ.Thefollowingisanaturalcorollarywhichis particularlyrelevantfor probabilitymeasures μwith boundeddistributionf and mean valueabouttheorigin.

Corollary2.Let m,n∈N_{,μ, Ψ, MΨ be asin Theorem 1 and letthe densityf of μ be}

boundedabove by aconstant K >0. Furthermore,letT be thesmallestpositive integer suchthat

μ([₋T, T)mn)_≥1₋δ/2. (21)

Then, forany δ∈(0,1),inequality(5)holds with

κ:= 1 2min

1

MΨ,

_δ

2K(2T)mn_ΣΨ 1/m

. (22)

Proof. With respect to Theorem 2, let p = ∞ and A be the collection of cubes P∆

thatexactlytiles[₋T,T)mn_{. Then,#A= (2T)}mn _{andthus Σ}

f ≤K(2T)mn. Now,(22)

triviallyfollowsfrom(20). 2

2.3. Numericalexamples

Inwhatfollows,we willusethestandardGaussian errorfunction

erf(x) :=√1

2π x

−∞

e−t2/2dt .

Itisreadilyverifiedthatthefunctionerf is continuous,strictlyincreasing andthat

lim

x→−∞erf(x) = 0 and x→lim+∞erf(x) = 1.

Asusual,for0< y <1,defineerf−1(y) tobetheuniquevaluex_∈R_{suchthaterf(x)=y.}

Furthermore,defineformally erf−1(0):=_−∞anderf−1(1):= +_∞.

κ= δ

√

2π

8·N·ΣΨ

, (23)

where N :=_⌈erf−1(1₋δ/4)_⌉.Here_⌈x_⌉isthe“ceiling”ofx,thatisthesmallestinteger thatisbiggerthanorequaltox∈R_{.Wenowconsiderexplicitapproximatingfunctions.}

First,letΨ bethefunctiongivenbyΨ(q)= 0 ifq_≤0,

Ψ(q) := 1

2q·log2q if q≥2 and Ψ(1) := 1/2.

Then ΣΨ<1.555 andonmakinguse of(23)we obtainthefollowing tablefor valuesof

N andκ:

δ 0.5 0.25 0.1 0.01 10−3 ₁₀−5

N 2 2 3 4 4 5

κ 0.05 0.025 0.0067 5·10−4 5·10−5 4·10−7

It follows for instance from this set of data that for 99% ofthe values of the random variable xwith normaldistribution_N(0,1),onehasthat

qx > 1

2000·Ψ(q) for all q∈N.

Inthenextexample,wefixaQ∈N_{andconsidertheapproximatingfunctionΨ given}

by

Ψ(q) :=

₁

Q if 1≤q≤Q,

0 otherwise.

Then ΣΨ = 1 andonecanreadilyverifythat

(i) for at least 75% of the values of the random variable x with normal distribution

N(0,1),onehasthat

qx> 1

13Q for all q∈[−Q, Q], q= 0,

(ii) for at least 90% of the values of the random variable x with normal distribution

N(0,1),onehasthat

qx> 1

3. Diophantineapproximation onmanifolds

Theaim is to establishananalogue ofTheorem 2 for submanifoldsMof Rn_{. More}

precisely,weconsider theset_Bn(Ψ,κ)_{∩ M},where

Bn(Ψ, κ) :=_B1,n(Ψ, κ).

Thefactthatthepointsofinterestareofdependentvariables,reflectingthefactthatthey lieon_M, introducesmajor difficultiesinattemptingto describethemeasure theoretic structureof_Bn(Ψ,κ)_{∩ M}.

Non-degenerate manifolds. In order to make any reasonable progress with the above problemsitisnotunreasonabletoassumethatthemanifolds_Munderconsiderationare non-degenerate.Essentially, theseare smoothsubmanifoldsofRn _{whichare sufficiently}

curved so as to deviate from any hyperplane. Formally, amanifold _M of dimension d

embedded in Rn _{is said to be non-degenerate if it arises from a non-degenerate map} f : U _→ Rn _{where U is an open subset of} Rd _{and M := f(U). The map f : U →}

Rn_{,x→f(x)= (f1(x),. . . ,fn(x)) issaidtobel-non-degenerateat x∈U,where l∈}N_,

iff isltimescontinuouslydifferentiableonsomesufficientlysmallballcentredatxand thepartialderivativesoff atxofordersuptol spanRn_{.Themapf isnon-degenerate}

atx ifitis l-non-degenerate atx forsomel_∈N_{.As iswell known,anyreal connected}

analyticmanifoldnotcontainedinanyhyperplaneofRn_{isnon-degenerateateverypoint} [13].

ObservethatifthedimensionofthemanifoldMisstrictly lessthannthenwehave that|Bn(Ψ,κ)∩M|n = 0 irrespectiveoftheapproximatingfunctionΨ andκ.Thus,when referringtotheLebesguemeasureoftheset_Bn(Ψ,κ)_{∩ M}itisalwayswithreferenceto theinduced Lebesgue measure on_M. More generally,given asubset S of _M we shall write _|S_|M for themeasure ofS withrespect to theinducedLebesgue measureon _M. Withoutlossofgenerality,wewillassumethat_|M|M= 1 asotherwisethemeasurecan bere-normalised accordingly.

ThefollowingstatementisastraightforwardconsequenceofthemainresultofBernik, KleinbockandMargulis in[8].

Theorem BKM.Let M be a non-degenerate submanifold of Rn_{. Let Ψ :} Zn _→ R_{+ be}

monotonicallydecreasingin each variableandsuchthat

ΣΨ:=

q_∈Zn_\{0}

Ψ(q)<∞. (24)

Then,foranyδ∈(0,1),thereisaconstant κ>0dependingonM,ΣΨ andδonlysuch

that

Remark 2.Theorem BKM holds for arbitrary probability measures supported on M

that are absolutely continuous with respect to the induced Lebesgue measure on _M, thus givingan analogueof Theorem 2 formanifolds. As inthecase ofTheorem 1, the moregeneralresultfollowsfromtheLebesguestatement.

It is worth pointing out that the main resultin [8] actually implies that the union

κ>0Bn(Ψ,κ)∩ M has full measure on M. Theorem BKM as stated above follows

from[8,Theorem 1.1]2_{onusingthecontinuityofmeasures.Ourmaingoalistoquantify}

the dependenceof κonδ.Theorem 6of §5belowexplicitlyquantifiesthis dependence. However,thestatementisrathertechnicalandweprefertostatefornowacleanerresult thatshowsthatthedependencybetweenκandδispolynomial.

Theorem 3.Let l _∈N _{and let M be a compactd-dimensional C}l+1 _{submanifold of} Rn

that is l-non-degenerate at every point. Let μ be a probability measure supported on

M absolutely continuous with respect to _|._|M. Let Ψ : Zn _→ R_{+ be a monotonically}

decreasing functionin each variable satisfying (24).Then there existpositive constants

κ0,C1 dependingonΨandMonlyandC0 dependingonthedimensionof Monly such

that forany 0< δ <1,theinequality

μ(_Bn(Ψ, κ)_{∩ M})_≥1₋δ (26)

holds with

κ:= minκ0, C0Σ−_Ψ1δ, C1δd(n+1)(2l−1) . (27)

4. PreliminariesforTheorem 3

4.1. Localisationandparameterisation

Since _Misnon-degenerate everywhere,wecanrestrictourselfto consideringa suffi-ciently smallneighbourhoodofanarbitrarypointon_M.Bycompactness,_Mthencan be coveredwithafinitesubcollectionofsuchneighbourhoods.Therefore,inviewofthe finitenessofthecover,theexistenceofκ0,C0andC1satisfyingTheorem 3globallywill

followfromtheexistenceoftheseparametersforeveryneighbourhoodinthefinitecover:

κ0,C0and C1 shouldbetakentobe theminimumoftheirlocalvalues.

NowaswecanworkwithMlocally,wecanparameterise itwithsomemapf :U→Rn

definedonaballU inRd_{, whered= dimM. Notethatf mustbe atleast C}2_inorder

to ensurethat_Misnon-degenerate. Withoutloss ofgeneralityweassumethat

M=_{f(x) :x_∈U_}.

2 _{Throughout, results and page numbers within [8] are with reference to the arXiv version: math/}

Furthermore, using the Implicit Function Theorem if necessary, we can make f to be a Monge parametrisation, that is f(x) = (x1,. . . ,xd,fd+1(x),. . . ,fn(x)), where x =

(x1,. . . ,xd).Note thatf canbe assumed to be bi-LipschitzonU.This readilyfollows

fromthefactthatf isC1 _{butpossiblyrequiresafurthershrinkingofU.}

Let _Bn(Ψ,κ,_M) denote the orthogonal projection of _Bn(Ψ,κ)_{∩ M} onto the set of parametersU.Thus,

Bn(Ψ, κ,_M) :=_{x_∈U:a.f(x)> κΨ(a) for all a_∈Zn_{, a=0}. (28)}

The set _Bn(Ψ,κ)_{∩ M} and its projection _Bn(Ψ,κ,_M) are related by the bi-Lipschitz map f. Since bi-Lipschitzmaps only affect the Lebesgue measure of aset by a multi-plicativeconstant(in this casetheconstantwill depend onf only), itsufficesto prove

Theorem 3fortheprojectset.Moreprecisely, Theorem 3isequivalentto showingthat there existpositiveconstants κ0,C0 andC1 >0 depending onΨ andf onlysuchthat

forany0< δ <1,

|Bn(Ψ, κ,M)|d ≥(1−δ)|U_|d (29)

holdswithκgivenby(27).

4.2. Auxiliarystatements

WewilldenotethestandardL1(resp.Euclidean,infinity)normonRdby.1(resp. .₂,._∞).Alsoasbefore,givenanx_∈R_{,xwilldenotethedistanceofxfromthe}

nearestinteger.ThenotationB(x,r) willrefertotheEuclideanopenballofradiusr >0 centred atx and Sd−1 _{willdenote the unitsphereindimension d≥1 (withrespect to}

theEuclideannorm).Furthermore, throughout

Vd :=

πd/2

Γ (1 +d/2)

is the volume of the d-dimensional unit ball and Nd denotes the Besicovitch covering

constant.

Remark3.ForfurtherdetailsontheBesicovitchcoveringconstant,cf.[9].Wewillonly needinwhatfollowstheinequalityNd≤5d satisfiedbythisconstant.

The proof of Theorem 3 involves two separate cases that take into consideration the relative size of the gradient of f(x)_·q, where q = (q1,. . . ,qn) ∈ Zn \ {0} and

f(x)_·q=f1(x)q1+· · ·+fn(x)qnisthestandardinnerproduct off(x) andq.Thefirst

In whatfollows, ∂β will denote partial derivationwith respect to amulti-indexβ =

(β1,. . . ,βd) ∈ Nd0, where N0 will stand for the set of non-negative integers, that is

N_{0:={0,1,2,. . .}.Furthermore,|β|we willmeantheorderofderivation,thatis|β|:=} β1+· · ·+βd. Also, ∂ik will denote the differential operator corresponding to the kth

derivativewithrespect totheith variable,thatis, ∂k

i :=∂k/∂xki.

Theorem4.LetU_⊂Rd _{beaballofradiusrandf ∈C}2_{(2U),where2Uistheballwith}

thesame centreasU andradius2r.Let

L∗:= sup

|β|=2,x∈2U

∂βf(x)∞ and L:= max

L∗, 1

4r2

. (30)

Then, forevery δ′_{>0andevery q∈}Zn_{\ {0},theset of x∈U suchthat}

f(x)·q< δ′

and

∇f(x)q_∞≥ ndLq_∞ (31)

has measureatmost Kdδ′|U|d,where∇f(x)q isthegradientof f(x)·q and

Kd:=

42d+1_dd/2_N d Vd

(32)

is aconstantdepending on donly.

Proof. Theproof of Theorem 4 follows onappropriately applying [8, Lemma 2.2]. For conveniencewerefertothislemmaasL2.2.WetakeMinL2.2tobeequaltothequantity

ndL,whereLisdefinedby(30).WesetδinL2.2tobeequaltoδ′_{appearinginTheorem 4.}

Then, inviewof (30) and thefact thatn,d,q_{∞ ≥}1,it follows thatthe hypotheses of L2.2(namely [8,Eq. (2.1a)&Eq. (2.1b)])aresatisfied.Thus, theconclusionofL2.2 implies Theorem 4 with theconstantCd inL2.2equalto Kd appearinginTheorem 4.

TheexplicitvalueofKd iscalculatedby‘tracking’thevaluesoftheauxiliaryconstants C′

d,Cd′′ andCd′′′ appearingin[8].Namely,3

C_d′ = Vd

22d_dd/2, C

′′

d = 2d+2, Cd′′′ = C′′

d C′

d ,

3 _{TherearetwotyposintheproofofL2.2thatoneshouldbeawareofwhenverifyingthevaluesofthe}

constants givenhere.Onpage 6 line−2,theinclusionregardingU(x) isthewrongwayround,itshould readU(x)⊃B(x, ρ

4√d).Next,onpage 7line 11,intherightmosttermofthedisplayedsetofinequalities thequantityδismissing,itshouldreadC′′′

andthen

Kd= 2dCd′′′Nd =

2d_C′′

dNd C′

d

= 2

4d+2_dd/2_N d Vd ·

2

NextinTheorem 5belowweconsider thecaseof ‘smallgradient’.This isanexplicit versionof[8,Theorem 1.4]. Firstweintroduceauxiliaryconstants.

GivenaCl_{map f :U→R}n _{definedonaballUinR}d_{,thesupremumofs∈Rsuch}

thatfor anyx _∈U and anyv _∈Sn−1 _{there exists an integerk, 0< k≤l, and aunit}

vectoru_∈Sd−1 _satisfying

∂k_{(f ·v)} ∂uk (x)

≥s (33)

willbecalledthemeasureofl-non-degeneracyof(f,U) andwillbedenotedbys(l;f,U). Here and elsewhere for a unit vector u _∈ Sd−1_{, ∂}k_/∂uk _{will denote the derivative in}

directionuoforderk.

AsinTheorem 4, theradius oftheball Uwill be denotedbyr. Throughout,welet

x0denotethecentreofU.Also,givenarealnumberλ>0,weletλUdenotethescaled

ballofradiusλrandwiththesamecentrex0asthatofU.Withthisinmind,consider

theballs

U+:= 3d+1_U,

˜

U := 3n+1U,

˜

U+:= 3n+d+2U.

(34)

Fortechnical reasons,thatwillsoon becomeapparent,inorder to dealwiththe‘small gradient’casewemakethefollowingassumptiononthemapf :U_→Rn_.

Assumption1.The mapf = (f1,. . . ,fn)isan n-tupleof Cl+1 functions definedon the

closureof U˜+ _{which isl-non-degenerateeverywhere ontheclosure ofU}˜+_.

Remark.In view of the discussion of §4.1, there is no loss of generality in imposing Assumption 1withinthecontextofTheorem 3.

Wedenotebys0 themeasureofnon-degeneracyoff onU˜+.NotethatAssumption 1

ensuresthat

s0:=s(l;f,U˜+)>0. (35)

sup

x_∈U˜+ ! ! ! !

∂k_f ∂u1. . . ∂uk(

x)

! ! ! !

2

≤M , (36)

where ∂ui means differentiationindirection ui. Note thattheleft-hand sideof (33)is

thelengthoftheprojectionof∂k_f(x)/∂uk _{onthelinepassingthroughvandhenceitis}

nobiggerthanM.Thisimpliesthat

s0 ≤ M .

Without lossofgenerality, wewillassumethattheradiusrof theball Usatisfies

r≤min

" _s

0·σ(l, d)

2_·3n+d+2√_dM,

ηs0

4·107₃n+d+2_{dM l}l+2_{(l+ 1)!}

#

, (37)

where

η:= min

1 16,

_V

d

2d+2_{dl(l+ 1)}1/l₅d

d(2l−1)(2l−2)

(38)

and where

σ(l, d) := 1 23l(d−1)/2 ·φ

_√

2_·(2l)2+l(l−1)/2_·(l+ 1)!−1,2, l d−1

(39)

with thequantity φ(ω,B,k) definedas

φ(ω, B, k) := ω

k(k−1)/2

2√2_·Bk_{·(k+ 1)!} (40)

forany integerk_≥1 andany realnumbersω,B >0.

Furthermore, definethefollowingconstants determinedbyf andU:

ρ1:=

s0

4ll_{(l+ 1)!}√_d(2r)

l_{, (41)}

τ:= r

l_s 0

4ll_{(l+ 1)!},

and

ρ2:=

s0

4ll_{(l+ 1)! τ}

M

l−1 $_τ$_1−1/√₂%%2

& s0

4ll_(l+1)! $_τ

M %l2

+τ1 +√1

2

2· (42)

Finally,let

ρ:= ρ1ρ2

ρ2

1+ (ρ2+ 2M2)2·

Theorem 5.LetU⊂Rd _{bea balland f = (f1,. . . ,fn) be an n-tupleof C}l+1 _functions

satisfyingAssumption 1. Then,forany 0< δ′_{≤1,any n-tupleT= (T}

1,. . . ,Tn)ofreal

numbers_≥1andany K >0satisfying

δ′KT1· · · · ·Tn

maxi=1,...,nTi ≤

1, (44)

define theset A(δ′_,K,T)tobe

A(δ′, K,T) :=

⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩

x_∈U:_∃q_∈Zn_{\ {0}such that} ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩

f(x)_·q< δ′

∇f(x)q_∞< K |q_i|< Ti, i= 1, . . . , n

⎫ ⎪ ⎪ ⎬ ⎪ ⎪ ⎭

. (45)

Then

|A(δ′, K,T)_|d≤E√n+d+ 1_·ε1

1/d(2l−1) |U_|d,

where

ε1:= max

⎛ ⎜ ⎝δ′,

⎛

⎝δ′KT1· · · · ·Tn

max

1≤i≤nTi ⎞ ⎠

1

n+1⎞ ⎟

⎠ (46)

and

E:=C(n+ 1)(3d_N

d)n+1ρ−1/d(2l−1), (47)

inwhichρ isgivenby(43)andC istheconstantexplicitlygiven by(62)below.

At first glance the statement of Theorem 5 looks very similar to [8, Theorem 1.4]. Westress thatthekeydifference isthatinourstatement theconstants aremade fully explicit.TheproofofTheorem 5isratherinvolvedandwillbe thesubjectof§6.

5. AstrengtheningandproofofTheorem 3

Inviewofthediscussionof§4.1,Theorem 3willfollowimmediatelyonestablishinga strongerresult(Theorem 6below),which explicitlycharacterises the dependence onΨ and _Mof the constants κ0, C0 and C1 appearingwithin thestatementof Theorem 3.

Let

CΨ:= sup

q=(q1,...,qn)∈Zn\{0}

Ψ(q)

n )

i=1

q+_i , whereq_i+:= max_{1,_|q_i|}. (48)

It is awell known factthat,underthe assumption thatΨ is monotonically decreasing ineachvariable,relation(24)impliesthat0< CΨ<∞.Alsodefinetheconstant

Sn:=

t∈Zn

2−

t_∞

2d(2l−1)(n+1)_,

whichisclearlyfinite andpositiveas thesumconverges.

Theorem 6.LetU_⊂Rd _{be aballwhose radiussatisfies (37)andlet f = (f1,. . . ,fn)be}

an n-tupleofCl+1 _{functionssatisfying Assumption1. LetΣ}

Ψ,L,Kd andE begivenby

(24),(30),(32)and(47)respectivelyand let

K0=E

_√

n+d+ 1·CΨ2n−1/2 √

ndL 1

n+11/d(2l−1) .

Given any δ >0,let

κ:= min

1

CΨ2n−1/2√ndL

, δ

2KdΣΨ

,

_δ

2K0Sn

d(n+1)(2l−1)

.

Then

|Bn(Ψ, κ,_M)_|d≥(1₋δ)_|U_|d. (49)

Clearly theaboveisanexplicitversionofTheorem 3inthecasewhenμisLebesgue measure.TheargumentsgivenintheproofofTheorem 2areeasilyadaptedtodealwith thegeneralsituation.

5.1. Proofof Theorem 6modulo Theorem 5

Forκ>0 andanyq_∈Zn_,define

A(κ;q) :=_{x_∈U:f(x)_·q< κΨ(q) &(31)holds_}

and

Clearlyitsufficestoprovethat

q∈Zn_\{0}

A(κ;q)

_d ≤ δ

2|U|d and

q∈Zn_\{0}

Ac(κ;q)

_d≤ δ

2|U|d. (50)

ByTheorem 4withδ′_{=κΨ(q),weimmediatelyhavethat|A(κ;q)|d≤K}

dκΨ(q)|U|d.

Then,summingover allq_∈Zn_{\ {0}gives}

q∈Zn_\{0} A(κ;q)

d

≤KdΣΨκ|U|d≤

δ

2|U|d. (51)

Now to establishthe second inequalityin(50),given an n-tuple t= (t1,. . . ,tn)∈ Nn0,

definetheset

Act:=

q=(q1,...,qn)∈Zn\{0}

2ti_≤q+

i<2ti+1

Ac(κ;q), (52)

whereq+_i = max_{1,_|q_i|}.Observethat

q∈Zn

Ac(κ;q) =

t∈Nn

0

Act. (53)

By(48)andthemonotonicityofΨ ineachvariable,foreveryq= (q1,. . . ,qn)∈Zn\ {0}

satisfyingtheinequalities2ti ≤q_i+<2ti+1,wehavethat

Ψ(q)≤CΨ

_n )

i=1

q+_i −1

≤CΨ n )

i=1

2−ti =CΨ2−

n i=1ti

and

q_∞≤2maxiti+1_.

Nowlet

δ′_=κC

Ψ2−

n

i=1ti, K=√ndL2maxiti+1 and T_i= 2ti+1 (1≤i≤n).

Then,Ac

t iseasilyseentobecontainedinthesetA(δ′,K,T) definedwithinTheorem 5.

ClearlyT1,. . . ,Tn≥1 andK >0.Sinceκ< CΨ−1,wehavethat0< δ′ <1.Finally,(44)

issatisfied,since

δ′_KT

1· · · · ·Tn

maxi=1,...,nTi =

κCΨ2−

n

i=1ti√ndL2maxiti+1* i2ti+1

=κCΨ2

n√_ndL

2(maxiti+1)/2 =

κCΨ2n−1/2 √

ndL

2t∞/2 < 1, (54)

wherethelastinequalityfollowsfromthedefinitionofκ.Therefore,Theorem 5is appli-cable anditfollowsthat

|Act|d≤ |A(δ′, K,T)|d≤E _√

n+d+ 1_·ε1

1/d(2l−1) |U_|d,

where E is given by (47) and where, from (54), the definition of δ′ _{and the fact that}

κCΨ<1,

ε1= max

⎛ ⎝κCΨ2−

n i=1ti,

κCΨ2n−1 √

ndL

2t_∞/2

1

n+1⎞ ⎠=

κCΨ2n−1 √

ndL

2t_∞/2

1

n+1 .

Then, using(53)andsummingoverallt_∈Nn

0,wefindthat

q∈Zn

Ac(κ;q)

_d≤

t∈Nn

0 E

⎛

⎝√n+d+ 1_·

κCΨ2n−1√ndL

2t_∞/2

1

n+1⎞ ⎠

1/d(2l−1)

|U_|d

=K0Snκ1/d(n+1)(2l−1)|U|d ≤ δ

2|U|d,

where thelatter inequalityfollows from thedefinitionofκ. Thisestablishes thesecond inequalityin(50)andthuscompletes theproofofTheorem 6moduloTheorem 5.

6. ProofofTheorem 5

To establish Theorem 5, we will follow the basicstrategy set outinthe proof of [8, Theorem1.4].Westressthatnon-trivialmodificationsandadditionsarerequiredtomake theconstantsexplicit.Tobeginwith,westateasimplifiedformof[8,Theorem 6.2]and, to thisend,variousnotionsarenowintroduced.

Given a finite dimensional real vector space W, ν will denote a submultiplicative function onthe exterioralgebra+W; thatis, ν isacontinuous functionfrom +W to

R_{+ suchthat}

ν(tw) =|t|ν(w) and ν(u_∧w)≤ν(u)ν(w) (55)

for any t _∈R _{and u,w ∈}+_{W. Given a discrete subgroup Λ of W of rank k ≥1, let} ν(Λ) :=ν(v1∧ · · · ∧vk), where v1,. . . ,vk is abasis of Λ (this definition makes sense

from thefirstequation in(55)).Also,_L(Λ) willdenote theset ofallnon-zeroprimitive subgroups of Λ. Furthermore, given C,α > 0 and V _⊂ Rd_{, a function f : x ∈ V →} f(x)_∈R_{is saidto be(C,α)-good onV ifforany openballB ⊂V and anyǫ>0,}

|{x_∈B:_|f(x)_|< ǫsup

x∈B|

Theorem7.([8, Theorem 6.2]) LetW bead+n+ 1dimensional realvector spaceand Λ be a discrete subgroup of W of rank k. Let a ball B = B(x0,r) _⊂ Rd _{and a map} H : ˆB _→GL(W) be given, where Bˆ := 3k_{B.Take C,α >0, 0<ρ <˜ 1and let ν be a}

submultiplicativefunctionon +W.Assumethat forany Γ_{∈ L}(Λ),

(i) thefunctionx_→ν(H(x)Γ)is(C,α)-good onBˆ, (ii) thereexistsx_∈B suchthat ν(H(x)Γ)≥ρ˜,and (iii) forallx_∈Bˆ,#{Γ∈ L(Λ)|ν(H(x)Γ)<ρ˜}<∞.

Then, forany positiveε_≤ρ,onehas

|{x_∈B:_∃ v_∈Λ_{\ {}0_}such that ν(H(x)v)< ε_{}|d ≤} k(3dNd)kC

ε ρ

α

|B_|d. (56)

Theorem 5isnowdeducedfromTheorem 7inthefollowingmanner.Withrespect to theparametersappearinginTheorem 7,we let

W =Rn+d+1

and

ν_∗ be the submultiplicative function introduced in[8, §7].

There isnothingto gaininformallyrecalling thedefinitionofν_∗.Allweneed to know isthatν_∗ asgivenin[8]hastheproperty that

ν∗(w)≤ w2 ∀ w∈+W (57)

and that its restriction to W coincides with the Euclidean norm. Next, the discrete subgroupΛ appearinginTheorem 7isdefinedas

Λ :=

_p

0 q

∈Rn+d+1_:p∈Z_,q∈Zn

. (58)

Note thatit has rank k = n+ 1, therefore the ball Bˆ appearing in the statement of

Theorem 7coincides with the ball U˜ definedby (34). Finally,we let the map H send

x_∈U˜ totheproductofmatrices

H(x) :=DUx, (59)

where

Ux:=

_{1 0 f(x)}

0 Id ∇f(x)

0 0 In

and Dis thediagonalmatrix

D:= diagε1

δ′,

ε1

K, . . . , ε1

K , -. /

dtimes

, ε1 T1

, . . . , ε1 Tn

(61)

definedviatheconstantsδ′_{, K, T}

1,. . . ,Tn,and ε1appearinginTheorem 5.

With theabovechoice ofparameters, onusing (57), it iseasily verifiedthat theset

A(δ′_{,K,T) definedby(45)withinthecontextofTheorem 5iscontainedinthesetonthe}

left-handsideof(56)withε:=ε1√n+d+ 1.TheupshotisthatTheorem 5followsfrom

Theorem 7 onverifying conditions(i), (ii)and (iii) therein withappropriate constants

C, αand ρ.Withthisinmind,wenotethatcondition(iii) isalreadyestablishedin[8, §7]foranyρ˜≤1.In §7below,wewillverifytheremainingconditions(i)&(ii)withthe following explicitconstants:

C:=

_{d(n+ 2)(d(n+ 1) + 2)}

2

α/2

max (C₁∗,2Cd,l), (62)

where

C₁∗:= max

2M s0·σ(l, d)

, 2 d+2

Vd ·

dl(l+ 1)_· M

s0 ·

2ll_{+ 1} σ(l, d)

1/l

(63)

(here,σ(l,d) isthequantitydefinedin(39)),

Cd,l:= 2

d+1_{dl(l+ 1)}1/l Vd

, (64)

α:= 1

d(2l₋1) (65)

and ρ˜=ρas definedby(43)(notethatρ<1).ThiswillestablishTheorem 5.

7. Verifyingconditions(i)&(ii)ofTheorem 7

Unless stated otherwise, throughoutthis section, Λ will be the discretesubgroup of

Rn+d+1_{givenby(58)andΓ∈ L(Λ) willbeaprimitivesubgroupofΛ.Verifyingcondition}

(i)of Theorem 7isbasedontwoseparate cases:onewhentherankofΓ is oneand the other caseof rank≥2.

7.1. Rankone caseof condition(i)

Proposition1.LetU⊂Rd _{beaball,F ⊂ C}l+1$_U˜+%_{beafamilyof realvaluedfunctions}

andλandγ be positiverealnumbers suchthat:

(1) theset _{∇f : f _{∈ F}} iscompactin_Cl−1$_U˜+%_,

(2) sup

f∈F

sup

x_∈U˜+|

∂βf(x)| ≤λ foranymulti-indexβ ∈Nd0 with 1≤ |β| ≤l+ 1,

(3) inf

f∈F u∈supSd−1

sup

1≤k≤l

∂∂kufk(x0)

≥γ,where x0 isthecentreof U,

(4) γ·σ(l, d)

2·3n+d+2√_dλ ≥ r , where r is the radius of U as defined in (37) and σ(l,d) is

defined in(39).

Then, forany f _{∈ F},wehave that

(a) f is$C1,_dl1%-goodonU˜,

(b) ∇f_∞ isC1,_d(l1₋₁₎

-goodonU˜,

where

C1=C1(γ, λ) := max

2λ γ·σ(l, d),

2d+2

Vd

dl(l+ 1)λ

γ

_2ll_{+ 1}

σ(l, d)

1/l

. (66)

Remark.Hypothesis (2) isadditionalto those madein[8,Proposition3.4]. Inshort,it isthis“extra”hypothesisthatyields anexplicitformulafortheconstantC1.Notethat

bythedefinitionofC∗

1 asgiven by(63),wehavethat

C₁∗=C1(s0, M).

UsingtheexplicitconstantC1appearinginProposition 1,itispossibletoadaptthe

proofof[8,Corollary3.5] togivethefollowingstatement.

Corollary3.Let U_⊂Rd _{bea ballandf = (f}

1,. . . ,fn) bean n-tuple of Cl+1 functions

satisfying Assumption1. Withreference toProposition 1,let

γ:=s0 and λ:=M.

Then, forany linear combinationf =c0+n_i=1cifi withc0,. . . ,cn∈R,we havethat

(a) f is$C∗

1,dl1 %

-goodonU˜,

(b) _∇f_∞ isC∗

1,d(l1−1)

-goodonU˜.