An Inhomogeneous Transference Principle and Diophantine Approximation
aa r X i v : . [ m a t h . N T ] F e b An Inhomogeneous Transference Principleand Diophantine Approximation
Victor Beresnevich ∗ York
Sanju Velani † York
Dedicated to Vasili Bernik – spasibo Basil!
Abstract
In a landmark paper [29], D.Y. Kleinbock and G.A. Margulis established the fun-damental Baker-Sprindzuk conjecture on homogeneous Diophantine approximationon manifolds. Subsequently, there has been dramatic progress in this area of research.However, the techniques developed to date do not seem to be applicable to inhomo-geneous approximation. Consequently, the theory of inhomogeneous Diophantineapproximation on manifolds remains essentially non-existent.In this paper we develop an approach that enables us to transfer homogeneousstatements to inhomogeneous ones. This is rather surprising as the inhomogeneoustheory contains the homogeneous theory and so is more general. As a consequence,we establish the inhomogeneous analogue of the Baker-Sprindzuk conjecture. Fur-thermore, we prove a complete inhomogeneous version of the profound theorem ofKleinbock, Lindenstrauss & Weiss [27] on the extremality of friendly measures. Theresults obtained in this paper constitute the first step towards developing a coherentinhomogeneous theory for manifolds in line with the homogeneous theory. ∗ EPSRC Advanced Research Fellow, grant EP/C54076X/1 † Research supported by EPSRC grant EP/E061613/1 and INTAS grant 03-51-5070 Introduction
The metrical theory of Diophantine approximation on manifolds dates back to the nineteenthirties with a conjecture of K. Mahler [31] in transcendence theory. The conjecture waseasily seen to be equivalent to a metrical Diophantine approximation problem restrictedto the Veronese curves V n := { ( x, . . . , x n ), x ∈ R } . Mahler’s conjecture remained a keyopen problem in metric number theory for over 30 years and was eventually solved bySprindzuk [37]. Moreover, its solution led Sprindzuk [39] to make an important generalconjecture which we shall shortly describe. The conjecture has been established by Klein-bock & Margulis in their landmark paper [29]. The main result of this paper establishesa complete inhomogeneous version of the theorem of Kleinbock & Margulis and indeed itsgeneralisation to friendly measures [27]. In order to describe these fundamental conjecturesand results it is convenient to introduce the notion of Diophantine exponents. Let m, n ∈ N and R m × n be the set of all m × n real matrices. Given X ∈ R m × n and θ ∈ R m , let w ( X , θ ) be the supremum of w > Q > q = ( q , . . . , q n ) ∈ Z n r { } satisfying k X q + θ k m < Q − w and | q | n Q , (1)where | q | := max {| q | , . . . , | q n |} is the supremum norm and k · k is the distance to thenearest integer point. Here and elsewhere q ∈ Z n and θ ∈ R m are treated as columns. Itfollows that whenever w ( X , θ ) is finite, the inequality k X q + θ k m < | q | − wn has infinitely many solutions q ∈ Z n if w < w ( X , θ ) and has at most finitely many solutions q ∈ Z n if w > w ( X , θ ). Further, let w × ( X , θ ) be the supremum of w > Q > q = ( q , . . . , q n ) ∈ Z n r { } satisfying Q h X q + θ i < Q − w and Q + ( q ) Q , (2)where Q y := Q ( y ) = m Y j =1 | y j | and Q + ( q ) := n Y i =1 max { , | q i |} for y = ( y , . . . , y m ). Also h y i denotes the unique point in [ − / , / m congruent to y ∈ R m modulo Z m . Thus, k · k = |h · i| . It follows that whenever w × ( X , θ ) is finite, theinequality Q h X q + θ i < Q + ( q ) − w has infinitely many solutions q ∈ Z n if w < w × ( X , θ ) and has at most finitely manysolutions q ∈ Z n if w > w × ( X , θ ). 2he homogeneous theory of Diophantine approximation corresponds to the special caseof θ = in the above inequalities. In this case the associated homogeneous exponentswill be denoted by w ( X ) := w ( X , ) and w × ( X ) := w ∗ ( X , ). A trivial consequence ofDirichlet’s theorem [36], or simply the ‘pigeon-hole principle’, is that w ( X ) > X ∈ R m × n . (3)Also it is readily seen that (1) implies (2) and therefore w × ( X , θ ) > w ( X , θ ) for all X ∈ R m × n and all θ ∈ R m . (4)The Diophantine exponents can in principle be infinite. Nevertheless, a relativelystraightforward consequence of the Borel-Cantelli lemma from probability theory is that(3) is reversed for almost all X ∈ R m × n with respect to Lebesgue measure on R m × n andmoreover that w × ( X ) = 1 for almost all X ∈ R m × n . (5)For completeness, we mention that X ∈ R m × n is said to be very well approximable (see[29, 36]) if w ( X ) > w × ( X ) > R m × n . Sprindzuk [39] conjectured that (5) remains true when X is restricted to any analyticnon-degenerate submanifold M of R n identified with either columns R n × ( simultaneousDiophantine approximation ) or rows R × n ( dual Diophantine approximation ) with respectto the Riemannian measure on M . This conjecture had been previously stated by A. Baker[3] for Veronese curves V n := { ( x, . . . , x n ), x ∈ R } . Essentially, non-degenerate manifoldsare smooth submanifolds of R n which are sufficiently curved so that they deviate fromany hyperplane with a ‘power law’ [4]. For the formal definition see [29]. Any real,connected analytic manifold not contained in a hyperplane of R n is non-degenerate. For aplanar curve, the non-degeneracy condition is simply equivalent to the condition that thecurvature is non-vanishing almost everywhere. Baker-Sprindzuk conjecture.
For any analytic non-degenerate submanifold M of R n w × ( X ) = 1 for almost all X ∈ M . (6)Note that in view of (3) and (4), the Baker-Sprindzuk conjecture implies that w ( X ) = 1 for almost all X ∈ M . (7)3n fact, this weaker statement also appears as a formal conjecture in [39]. In the case of theVeronese curves V n , (7) reduces to Mahler’s problem [31] and statement (6) reduces to thespecific conjecture of A. Baker mentioned above. Manifolds that satisfy (7) are referredto as extremal and manifolds that satisfy (6) are referred to as strongly extremal . To beprecise, either notion of extremality actually takes on two forms depending on whether R n is identified with R n × or R × n . However, by Khintchine’s transference principle bothforms are equivalent and it is pointless to distinguish between them. A priori, this is notthe case when considering ‘extremality’ within the inhomogeneous setting – see § R . The actual conjecturehad only been established for the Veronese curves V n with n n = 4 by Bernik & Borbat [12]. In their ground breaking work, Kleinbock & Margulis [29]established the Baker-Sprindzuk conjecture in full generality and moreover removed the‘analytic’ assumption. Theorem KM
Any non-degenerate submanifold of R n is strongly extremal. The work of Kleinbock & Margulis has led to various generalisations of the Baker-Sprindzuk conjecture. Kleinbock [25] has established that non-degenerate submanifolds ofstrongly extremal affine subspaces of R n are strongly extremal. He has also shown that non-degenerate complex analytic manifolds are strongly extremal [26]. Kleinbock & Tomanov[28] have generalised Theorem KM to the S -arithmetic setting. Kleinbock, Lindenstrauss& Weiss [27] have revolutionised the notion of extremality by introducing the concept ofmeasures being extremal rather than sets. Let µ be a measure supported on a subset of R m × n . We say that µ is extremal if w ( X ) = 1 for µ -almost every point X ∈ R m × n . Inother words, the set of X ∈ R m × n for which w ( X ) > µ -measure zero. We say that µ is strongly extremal if w × ( X ) = 1 for µ -almost every point X ∈ R m × n . Furthermore,if µ is a measure on R n then µ is (strongly) extremal if it is (strongly) extremal throughthe identification of R n with either R n × or R × n . In view of Khintchine’s transferenceprinciple, there is no difference which representation of R n is taken.The following constitutes the main result of Kleinbock, Lindenstrauss & Weiss [27]. Theorem KLW
Any friendly measure on R n is strongly extremal. The definition of friendly measures is given in §
2. At this point it suffices to say thatfriendly measures form a large and natural class of measures on R n including Rieman-nian measures supported on non-degenerate manifolds, fractal measures supported on self-similar sets satisfying the open set condition (e.g. regular Cantor sets, Koch snowflake,Sierpinski gasket) and conformal measures supported on limit sets of Kleinian groups. Inview of the former we have thatTheorem KLW = ⇒ Theorem KM.4 .3 Inhomogeneous theory
The central goal of this paper is to establish the inhomogeneous analogue of the Baker-Sprindzuk conjecture. Naturally, we begin by introducing the notion of extremality in theinhomogeneous theory of Diophantine approximation.
Definition 1 (Measures on R m × n ) Let µ be a measure supported on a subset of R m × n .We say that µ is inhomogeneously extremal if for all θ ∈ R m w ( X , θ ) = 1 for µ -almost all X ∈ R m × n . (8)We say that µ is inhomogeneously strongly extremal if for all θ ∈ R m w × ( X , θ ) = 1 for µ -almost all X ∈ R m × n . (9)These notions of extremality naturally generalise the homogenous ones which only require(8) and (9) to hold for θ = . A remark regarding the use of the word strongly in thedefinition of ‘inhomogeneously strongly extremal’ is in order. In the homogeneous case, (3)and (4) with θ = show that strong extremality implies extremality – exactly as one wouldexpect. In the inhomogeneous case there is no analogue of (3) and it is not at all obviousthat strong extremality implies extremality. However, the following result established in § Proposition 1
Let µ be a measure on R m × n . Then µ is inhomogeneously strongly extremal = ⇒ µ is inhomogeneously extremal . As already mentioned, there are two different forms of Diophantine approximation whenapproximating points in R n depending on whether R n is identified with R n × or R × n .The identification with the former corresponds to the simultaneous form and the lattercorresponds to the dual form. As a consequence of Khintchine’s transference principle, thetwo forms of approximation lead to equivalent notions of extremality in the homogeneouscase. However, Khintchine’s transference principle is not applicable in the inhomogeneouscase and the simultaneous and dual forms of extremality are not necessarily equivalent.Consequently, the two forms of extremality need to be considered separately. Definition 2 (Measures on R n ) Let µ be a measure supported on a subset of R n . If µ is inhomogeneously (strongly) extremal on R × n we say that µ is dually inhomogeneously(strongly) extremal . If µ is inhomogeneously (strongly) extremal on R n × we say that µ is simultaneously inhomogeneously (strongly) extremal . If µ is both dually and simultane-ously inhomogeneously (strongly) extremal then we simply say that µ is inhomogeneously(strongly) extremal . 5aturally, a manifold M ⊂ R n is called inhomogeneously (strongly) extremal if theRiemannian measure on M is inhomogeneously (strongly) extremal. The Veronese curves V n have been shown to be dually inhomogeneously extremal in the real [13], complex[41], p -adic [14, 40] and ‘mixed’ [17] cases. These results are natural generalisations ofMahler’s conjecture to the inhomogeneous setting. Most recently, Badziahin [1] has ex-tended Schmidt’s homogeneous result [34] by showing that non-degenerate planar curvesare dually inhomogeneously extremal. Strikingly this constitutes the only known resultbeyond the Veronese curves. However, one would expect that an inhomogeneous analogueof Theorem KM holds in full generality. Inhomogeneous Baker-Sprindzuk conjecture.
Any non-degenerate submanifold of R n is inhomogeneously strongly extremal. In view of Proposition 1, this implies the weaker conjecture that any non-degenerate sub-manifold M of R n is inhomogeneously extremal . As discussed above the weaker conjectureis known to be true for Veronese curves. Regarding the stronger conjecture nothing isknown. Nevertheless, given Theorem KLW, it is natural to broaden the conjecture tofriendly measures. Conjecture.
Any friendly measure on R n is inhomogeneously strongly extremal. Let µ be a non-atomic, locally finite, Borel measure on R m × n . Obviously, we have that µ is inhomogeneously (strongly) extremal = ⇒ µ is (strongly) extremal.The right hand side corresponds to the special choice of θ = in the definition of in-homogeneously (strongly) extremal. In this paper we show that the above implication isreversed for a large class of measures. Theorem 1
Let µ be a measure on R m × n . (A) If µ is contracting almost everywhere then µ is extremal ⇐⇒ µ is inhomogeneously extremal. (B) If µ is strongly contracting almost everywhere then µ is strongly extremal ⇐⇒ µ is inhomogeneously strongly extremal. So as to avoid introducing various technical notions at this point, the definition ofcontracting measures is postponed to the next section. It suffices to say that friendlymeasures on R n fall within the class of strongly contracting measures. Thus, Theorem 1B(part (B) of Theorem 1) together with Theorem KLW establishes the conjecture statedabove for friendly measures. 6 heorem 2 Any friendly measure on R n is inhomogeneously strongly extremal. Riemannian measures supported on non-degenerate manifolds are know to be friendly[27]. Thus, Theorem 2 gives a complete inhomogeneous analogue of Theorem KM andthereby settles the inhomogeneous Baker-Sprindzuk conjecture.
Theorem 3
Any non-degenerate submanifold of R n is inhomogeneously strongly extremal. It is worth mentioning that the class of contracting measures is not limited to friendlymeasures. To illustrate this we restrict our attention to simultaneous Diophantine approx-imation. Thus, R n is identified with R n × . In § M of R n falls within the class of contractingmeasures and indeed within the class of strongly contracting measure if a mild condition isimposed on M . However, affine subspaces of R n are differentiable manifolds and for obvi-ous reasons they do not support friendly measures. Specializing Theorem 1 to differentiablemanifolds gives the following statement. Theorem 4
Let M be a differentiable submanifold of R n . Then (A) M is extremal ⇐⇒ M is simultaneously inhomogeneously extremal .Furthermore, suppose that at almost every point on M the tangent plane is not orthogonalto any of the coordinate axes. Then (B) M is strongly extremal ⇐⇒ M is simultaneously inhomogeneously strongly extremal. Recall, that the left hand side of the above implications are homogeneous statementsand the notions of simultaneously and dually (strongly) extremal coincide. Examples of(strongly) extremal differentiable submanifolds that do not fall within the remit of Theorem2 are given in [6, 25, 34]. Thus, Theorem 4 is not vacuous.The following diagram summarizes the connections between the various notions of ex-tremality for strongly contracting measures on R m × n . µ is extremal Theorem 1A ⇐⇒ µ is inhomogeneously extremal(L) ⇑ ⇑ (R) µ is strongly extremal Theorem 1B ⇐⇒ µ is inhomogeneously strongly extremalAs mentioned in § §
5, we develop an abstract framework within which we establish a general inhomogeneoustransference principle – namely Theorem 5. The key step in establishing Theorem 1 followsas an application of this inhomogeneous transference principle.
Remark.
A direct and self-contained proof of Theorem 4A can be found in our recentarticle [10]. The main motivation behind [10] is to foreground and significantly simplify thekey ideas involved in establishing the inhomogeneous transference principle of §
5. Indeed,anyone interested in the proof of Theorem 5 and thus Theorem 1 may find it useful firstto look at [10].
In this section we start by formally introducing the class of ‘contracting’ measures alludedto in Theorem 1 above. We then show that friendly measures are contracting and thatthe Riemannian measure on a differentiable submanifold of R n × is contracting. Thisestablishes Theorems 2 and 4 from Theorem 1. We begin by recalling some standard notions. If B is a ball in a metric space Ω then cB denotes the ball with the same centre as B and radius c times the radius of B . A measure µ on Ω is non-atomic if the measure of any point in Ω is zero. The support of µ is thesmallest closed set S such µ (Ω \ S ) = 0. Also, recall that µ is doubling if there is a constant λ > B with centre in S µ (cid:0) B (cid:1) λ µ (cid:0) B (cid:1) . (10)The class of contracting measures µ is defined via the behavior of µ near planes in R m × n .More precisely, the planes are given by L a , b := { X ∈ R m × n : X a + b = } with a ∈ R n , | a | = 1 and b ∈ R m , (11)where | · | is the Euclidean norm. Given ε = ( ε , . . . , ε m ) ∈ (0 , + ∞ ) m , the ε -neighborhoodof the plane L a , b is given by L ( ε ) a , b := { X ∈ R m × n : | X j a + b j | < ε j ∀ j = 1 , . . . , m } , (12)where X j is the j -th row of X . In the case that ε = · · · = ε m = ε , we simply write L ( ε ) a , b for the symmetric ε -neighborhood of L a , b . 8 efinition 3 A non-atomic, finite, doubling Borel measure µ on R m × n is strongly con-tracting if there exist positive constants C , α and r such that for any plane L a , b , any ε = ( ε , . . . , ε m ) ∈ (0 , + ∞ ) m with min { ε j : 1 j m } < r and any δ ∈ (0 ,
1) thefollowing property is satisfied: for all X ∈ L ( δ ε ) a , b ∩ S there is an open ball B centred at X such that B ∩ S ⊂ L ( ε ) a , b (13)and µ (5 B ∩ L ( δ ε ) a , b ) Cδ α µ (5 B ) . (14)The measure µ is said to be contracting if the property holds with ε = · · · = ε m = ε . Remark.
The property given by (13) and (14) indicates the rate at which the µ -measureof the ε -neighborhood of L a , b decreases when contracted by the multiplicative factor δ .Also, note that µ is strongly contracting = ⇒ µ is contracting.The definition of (strongly) contracting is in essence a global statement – the ‘property’is required to hold for all X in the support S . However, with the view of establishing‘extremality’ results such as Theorem 1, sets of µ -measure zero are irrelevant and thenotion of (strongly) contracting almost everywhere suffices. Formally, we say that µ is (strongly) contracting almost everywhere if for µ -almost every point X ∈ R m × n there is aneighborhood U of X such that the restriction µ | U of µ to U is (strongly) contracting. The notion of friendly measures introduced in [27] identifies purely geometric conditionson measures on R n that are sufficient to guarantee strong extremality. The class of friendlymeasures is defined via the behavior of µ near hyperplanes L in R n .Let µ be a Borel measure on R n and as usual let S denote the support of µ . We saythat µ is non-planar if µ ( L ) = 0 for any hyperplane L . Furthermore, given L and a ball B with µ ( B ) >
0, let k d L k µ,B be the supremum of dist( x , L ) over x ∈ S ∩ B . Here dist( x , L )is the Euclidean distance of x from L . Next, let U be an open subset of R n . Given positivenumbers C and α , the measure µ is called ( C, α ) -decaying on U if for any non-empty openball B ⊂ U centred in S , any affine hyperplane L of R n and any ε > µ ( B ∩ L ( ε ) ) C (cid:18) ε k d L k µ,B (cid:19) α µ ( B ) . (15) Definition 4
A non-atomic, Borel measure µ on R n is called friendly if for µ -almost everypoint x ∈ R n there is a neighborhood U of x such that the restriction µ | U of µ to U isfinite, doubling, non-planar and ( C, α )-decaying for some positive C and α .9n the next two sections we shall establish that friendly measures on R n identified eitherwith R × n or R n × are strongly contracting. R m × Proposition 2
Any friendly measure µ on R m × is strongly contracting almost everywhere.Proof. Let µ be a friendly measure on R m identified with R m × . Then for µ -almost everypoint X ∈ R m × there is a neighborhood U of X such that µ | U is ( C, α )-decaying on U for some fixed C, α >
0. Without loss of generality we can assume that µ = µ | U . The factthat µ is non-planar, implies that there are n + 1 linearly independent points X , . . . , X n in the support S of µ . This ensures that there exists a real number r > r contains S . Also, note that since R m is identified with R m × , the set L a, b appearing in the definition of strongly contracting issimply a point. Thus, for any point L a, b the ε -neighborhood given by (12) n =1 is a rectanglewith sides of length 2 ε i ( i = 1 , . . . , m ). In particular, if min j m ε j < r then S
6⊂ L ( ε ) a, b . (16)Fix δ ∈ (0 ,
1) and take any point X ∈ S ∩ L ( δ ε ) a, b . Without loss of generality, we assume thatthis intersection is non-empty. The goal is to construct a ball B centred at X satisfying(13) and (14) in the definition of strongly contracting. To start with, let B ′ be an arbitraryball centred at X such that B ′ ⊂ L ( ε ) a, b . (17)This is possible as X ∈ L ( ε ) a, b and by definition L ( ε ) a, b is an open set. By (16) and (17), thereis a real number τ > τ B ′ ∩ S
6⊂ L ( ε ) a, b and τ B ′ ∩ S ⊂ L ( ε ) a, b . (18)By (18), there exists a point X ′ ∈ (cid:16) τ B ′ ∩ S (cid:17) \ L ( ε ) a, b . By the choice of X ′ , there exists a j ∈ { , . . . , m } such that (cid:12)(cid:12) X ′ j + b j (cid:12)(cid:12) > ε j . (19)Recall that X ′ j and b j are the j -th coordinates of X and b respectively. With reference to § L be given by X j + b j = 0 and B = 5 τ B ′ . It follows from (19) that k d L k µ,B > ε j . Since µ is ( C, α )-decaying, (15) with ε := δε j implies that µ (5 τ B ′ ∩ L ( δ ε ) a, b ) µ (5 τ B ′ ∩ L ( δε j ) ) < C (cid:18) δε j ε j (cid:19) α µ (5 τ B ′ ) = C δ α µ (5 τ B ′ ) . The upshot of this is that the ball τ B ′ satisfies conditions (13) and (14). The otherconditions of strongly contracting are trivially met and the proof is complete. ⊠ .2.2 Friendly measures on R × n Proposition 3
Any friendly measure µ on R × n is strongly contracting almost everywhere.Proof. Let µ be a friendly measure on R n identified with R × n . Then for µ -almost everypoint X ∈ R × n there is a neighborhood U of X such that µ | U is ( C, α )-decaying on U for some fixed constants C, α >
0. Without loss of generality we can assume that µ = µ | U . The fact that µ is non-planar, implies that there are n + 1 linearly independentpoints X , . . . , X n in the support S of µ . This ensures that there exists a real number r > ε ∈ (0 , r ) and any hyperplane L a ,b the ε -neighborhood given by(12) m =1 cannot contain all the points X , . . . , X n . It follows that for any hyperplane and0 < ε < r , we have that S
6⊂ L ( ε ) a ,b . (20)Fix δ ∈ (0 ,
1) and take any point X ∈ S ∩ L ( δ ε ) a, b – we may as well assume that thisintersection is non-empty. Now, let B ′ be an arbitrary ball centred at X such that B ′ ⊂ L ( ε ) a ,b . (21)This is possible as L ( ε ) a ,b is open. By (20) and (21), there is a real number τ > τ B ′ ∩ S
6⊂ L ( ε ) a ,b and τ B ′ ∩ S ⊂ L ( ε ) a ,b . (22)By (22), there exists a point X ′ ∈ τ B ′ ∩ S which is not contained in L ( ε ) a ,b . With referenceto § L = L a ,b and B = 5 τ B . It follows that k d L k µ,B > ε which together with (15)implies that µ (cid:0) τ B ′ ∩ L ( δε ) a ,b (cid:1) C δ α µ (cid:0) τ B ′ (cid:1) . Thus, the ball τ B ′ satisfies conditions (13) and (14) in the definition of strongly contracting.The other conditions of strongly contracting are trivially met and the proof is complete. ⊠ We begin by establishing Theorem 4B - part (B) of Theorem 4. Clearly, we only have toprove the necessity part as the right hand side of the statement contains the left hand side.Thus, we are given that the differentiable submanifold M of R n is strongly extremal. Let m denote the Riemannian measure on M . The aim is to show that m is inhomogeneouslystrongly extremal on R n × – see Definition 2 in § m as a measure on R n × is strongly contractingalmost everywhere.Take any point y ∈ M such that the tangent plane to M at y is not orthogonal toany of the coordinate axes. Since the latter property holds almost everywhere on M , it11uffices to prove Theorem 4B for a neighborhood P of y . Without loss of generality, wecan assume that there is a C (1) local parameterisation of P given by f : U → M . Here U is a ball in R d centred at x ∈ U such that f ( x ) = y and d = dim M . By the conditionon y imposed above, there is a direction v ∈ R d such that the tangent direction ∂ f ( x ) ∂ v isnot orthogonal to any of the coordinate axes. This means that there exists some κ > κ − < (cid:12)(cid:12)(cid:12)(cid:12) ∂f i ( x ) ∂ v (cid:12)(cid:12)(cid:12)(cid:12) < κ/ i n .Since f is C (1) , there exists a sufficiently small ball B ⊂ U centered at x such that κ − < (cid:12)(cid:12)(cid:12)(cid:12) ∂f i ( x ) ∂ v (cid:12)(cid:12)(cid:12)(cid:12) < κ for all 1 i n and all x ∈ B . (23)Without loss of generality, take f ( B ) to be the neighborhood P ⊂ M of y mentionedabove. We now slice B with respect to the direction v so as to reduce the problem athand to one concerning differentiable curves. Since M is strongly extremal and using thefact that sets of full measure are invariant under diffeomorphisms, the set E := { x ∈ B : w × ( f ( x )) = 1 } has full Lebesgue measure in B . Now for any x ′ ∈ R d orthogonal to v , consider the line L x ′ in R d given by L x ′ := { x = x v + x ′ ∈ R d : x ∈ R } . Also, let E x ′ := E ∩ L x ′ and B x ′ := B ∩ L x ′ . Clearly, B x ′ is either an interval or is empty and E x ′ ⊂ B x ′ . For obvious reasons, weonly consider the situation when B x ′ = ∅ . Since E has full measure in B , it follows fromFubini’s theorem that for almost every x ′ the slice E x ′ has full measure in B x ′ . Now let f x ′ denote the map f restricted to B x ′ . Clearly, f x ′ is a diffeomorphism from B x ′ onto thecurve M x ′ := f ( B x ′ ) . Since E x ′ has full measure in B x ′ and f x ′ is a diffeomorphism, M x ′ is strongly extremal foralmost all x ′ orthogonal to the direction v .Now we fix any x ′ ∈ R d orthogonal to v such that the curve C := M x ′ is non-empty andstrongly extremal. Define the map g = ( g , . . . , g n ) : I → R n from the interval I := { x ∈ R : x v + x ′ ∈ B } such that g ( x ) = f ( x v + x ′ ). By (23), we have that κ − | g ′ i ( x ) | κ for all 1 i n and all x ∈ I . (24)12et µ denote the induced Lebesgue measure on C and identify R n with R n × . The key partof the proof is to that show that µ as a measure on R n × is strongly contracting. Thisinvolves verifying (13) and (14). Regarding (14) we can assume that δ < / C >
2. Also, given that R n is being identified with R n × , theset L a, b appearing in the definition of strongly contracting is simply a point. Now choosea real number r > L a, b and ε = ( ε , . . . , ε n ) ∈ (0 , + ∞ ) n withmin j n ε j < r , we have that C 6⊂ L ( ε ) a, b . The latter is readily deduced from (24). In what follows we fix a point L a, b , a vector ε with 0 < min j n ε j < r and a δ ∈ (0 , / ε . . . ε n and that µ ( L ( δ ε ) a, b ∩ C ) = 0. We now verify that µ ( L ( δ ε ) a, b ∩ C ) √ nκ δε . (25)Let X and X ′ be any two points in L ( δ ε ) a, b ∩ C . Thus, X = g ( x ) and X ′ = g ( x ′ ) for some x, x ′ ∈ I . It follows that | g ( x ) − b /a | < δε and | g ( x ′ ) − b /a | < δε , (26)where b is the first coordinate of b associated with the point L a, b and g is the firstcoordinate function of g . By the Mean Value theorem, there exists some θ ∈ [0 ,
1] suchthat | ( g ( x ) − b /a ) − ( g ( x ′ ) − b /a ) | = | g ( x ) − g ( x ′ ) | = | x − x ′ | (cid:12)(cid:12) g ′ (cid:0) θx + (1 − θ ′ ) x ′ (cid:1)(cid:12)(cid:12) . (27)By (24), (26) and (27), it follows that | x − x ′ | κδε . (28)In view of (24), g i is monotonic for every i and therefore the set g − ( L ( δ ε ) a, b ∩ C ) ⊂ I is aninterval. Let x and x be the endpoints of this interval with x < x . Clearly, (28) is validwith x = x and x ′ = x . Therefore, µ ( L ( δ ε ) a, b ∩ C ) = Z x x | g ′ ( x ) | dx ( ) √ nκ | x − x | ( ) √ nκ κδε . This is precisely (25). Now, let B be a ball centred at X of radius ε /
2. By the choice of X and the fact that δ < /
2, we have that B ⊂ L ( ε ) a, b . (29)By the Mean Value Theorem and (24), for any x ∈ I ′ := { x ′ ∈ I : | x − x ′ | < ε / (2 √ nκ ) } we have that | g ( x ′ ) − g ( x ) | = | g ′ ( θ ′ x ′ + (1 − θ ′ ) x ) | | x − x ′ | < ε / .
13t follows that g ( x ′ ) ∈ B for any x ′ ∈ I ′ and that | I ′ | > ε / (2 √ nκ ). Hence, µ (5 B ∩ C ) > µ ( B ∩ C ) > Z I ′ | g ′ ( x ) | dx ( ) > κ √ n/ · | I ′ | > ε / . (30)On combining inequalities (25) and (30), we obtain that µ (5 B ∩ L ( δ ε ) a, b ∩ C ) µ ( L ( δ ε ) a, b ∩ C ) √ nκ δε √ nκ δ µ (5 B ∩ C ) . (31)Clearly, (29) verifies (13) and (31) verifies (14). Thus, the measure µ on R n × is stronglycontracting. By Theorem 1B, it follows that µ is inhomogeneously strongly extremalon R n × . By definition, µ or equivalent C is simultaneously inhomogeneously stronglyextremal. This establishes Theorem 4B in the case that M is a differentiable curve. Todeal with manifolds in general, we appeal to Fubini’s theorem. For any θ ∈ R n , considerthe sets E θ := { x ∈ B : w × ( f ( x ) , θ ) = 1 } and E θ x ′ := E θ ∩ L x ′ . Clearly, E θ x ′ ⊂ B x ′ . For almost every x ′ the measure µ on the corresponding curve M x ′ is simultaneously inhomogeneously strongly extremal. Thus, for almost every x ′ the slice E θ x ′ has full Lebesgue measure in B x ′ . Hence, by Fubini’s theorem we have that E θ has fullLebesgue measure in B . Consequently, f ( E θ ) has full Riemannian measure in P := f ( B ).This completes the proof of Theorem 4B. ⊠ The proof of Theorem 4A follows the same line of argument as above. However, weonly require that the inequality in (23) holds for at least one value of i rather than for all i . This is the case for any differentiable manifold irrespective of the direction v . Hencethere is no extra hypothesis on M in Theorem 4A. The details are left to the reader. Asmentioned at the end of § Given a measure µ on R m × n , suppose we are interested in establishing that µ is inhomo-geneously strongly extremal. Clearly, this would follow on showing that for all θ ∈ R m w × ( X , θ ) µ -almost all X ∈ R m × n and w × ( X , θ ) > µ -almost all X ∈ R m × n . (32)Establishing inhomogeneous extremality corresponds to similar statements with w × ( X , θ )replaced by w ( X , θ ). Note that as a consequence of (3) and (4), in the homogeneous case14 θ = ) the set of X satisfying w ( X , θ ) > w × ( X , θ ) > w ( X , θ ) > Proposition 4
Let µ be an extremal measure on R m × n . Then for all θ ∈ R m , w × ( X , θ ) > w ( X , θ ) > for µ -almost all X ∈ R m × n . If µ is extremal, then w ( X ) = 1 for µ -almost all X ∈ R m × n and Proposition 4 readilyfollows from (4) and the following statement. Lemma 1
Let X ∈ R m × n such that w ( X ) = 1 . Then for all θ ∈ R m , w ( X , θ ) > . (33)The proof of the lemma utilises basic ‘transference’ inequalities relating various formsof Diophantine exponents. These we briefly describe. A form of Khintchine’s transferenceprinciple due to Dyson [36, Theorem 5C] relates the homogeneous exponents of X and itstranspose t X . It states that w ( X ) = 1 ⇐⇒ w ( t X ) = 1 for all X ∈ R m × n . (34)In the spirit of Cassels [20, Chapter 5], Bugeaud & Laurent [19] have recently discoveredtransference inequalities that relate the Diophantine exponents w ( X , θ ) with their uniformcounterparts b w ( X , θ ). The latter are defined as followed. Given X ∈ R m × n and θ ∈ R m ,let b w ( X , θ ) be the supremum of w > Q there isa q ∈ Z n r { } satisfying (1). As with the standard non-uniform exponents, a trivialconsequence of Dirichlet’s theorem is that b w ( X ) > X ∈ R m × n . (35)Also, the following inequalities are easily verified. w ( X , θ ) > b w ( X , θ ) > . (36) Theorem BL (Bugeaud & Laurent)
Let X ∈ R m × n . Then for all θ ∈ R m , w ( X , θ ) > b w ( t X ) and b w ( X , θ ) > w ( t X ) (37) with equalities in ( ) for almost all θ ∈ R m .
15e are now fully armed to proceed with the proof of above lemma.
Proof of Lemma 1.
We are given that w ( X ) = 1. Hence, by (34) it follows that w ( t X ) = 1.This together with (35) and (36) θ = applied to t X implies that b w ( t X ) = 1. In turn, thiscombined with (37) implies that w ( X , θ ) > ⊠ Remark 1.
It is worth pointing out that Lemma 1, which allows us to deduce Proposition 4and thereby reduce the proof of Theorem 1 to establishing upper bounds for the associatedDiophantine exponents, can in fact be proved without appealing to Theorem BL. Indeed,a proof can be given which only makes use of classical transference inequalities; namelyTheorem VI of Chapter 5 in [20]. Thus, the proof of Proposition 4 and therefore Theorem1 is not actually dependent on the recent developments regarding transference inequalities.
Remark 2.
Theorem BL actually gives us information beyond Lemma 1. It enables usto deduce that inequality (33) is in fact an equality for almost all θ ∈ R m . Thus, the realsignificance of Theorem 1A is in establishing a global result which holds for all θ ∈ R m . Let µ be a measure on R m × n . Given that µ is inhomogeneously strongly extremal we wishto conclude that µ is inhomogeneously extremal. This as we shall now see is a simpleconsequence of (4) and Lemma 1.We are given that for any θ ∈ R m , w × ( X , θ ) = 1 for µ -almost all X ∈ R m × n . By(4), it follows that for any θ ∈ R m , w ( X , θ ) µ -almost all X ∈ R m × n . Thus, weonly need to show that for any θ ∈ R m , w ( X , θ ) > µ -almost all X ∈ R m × n . Since µ is inhomogeneously strongly extremal we trivially have that µ is strongly extremal andtherefore extremal. In other words, w ( X ) = 1 for almost all X ∈ R m × n . This togetherwith Lemma 1 yields the desired statement. ⊠ The goal of this section is to reformulate Theorem 1 so that the new statement can bededuced via the general framework developed in §
5. On the other hand, the reformulation isnatural even for a direct proof of Theorem 1 and thereby motivates the general framework.Theorem 1 consists of two parts which we refer to as Theorem 1A and Theorem 1B.We will concentrate on establishing Theorem 1B. The proof of Theorem 1A is similar inspirit and we shall indicate the necessary modifications that need to be made.With the intention of proving Theorem 1B, let µ be a strongly extremal measure on R m × n A θ m,n := { X ∈ R m × n : w × ( X , θ ) > } . In view of Proposition 4, Theorem 1B is reduced to showing that µ ( A θ m,n ) = 0 for all θ ∈ R m . (38)The key towards establishing (38) is the following reformulation. Let T denote a countablesubset of R m + n such that for every t = ( t , . . . , t m + n ) ∈ T m X j =1 t j = n X i =1 t m + i . (39)For t ∈ T , consider the diagonal unimodular transformation g t of R m + n given by g t := diag { t , . . . , t m , − t m +1 , . . . , − t m + n } . (40)For X ∈ R m × n , define the matrix M X := I m X I n ! , where I n and I m are respectively the n × n and m × m identity matrices. The matrix M X is a linear transformation of R m + n . Given θ ∈ R m , let M θX : a M θX a := M X a + Θ , where Θ := t ( θ , . . . , θ m , , . . . , ∈ R m + n . Thus, M θX is an affine transformation of R m + n .Let A = Z m × ( Z n r { } ) . (41)Then, for ε > t ∈ T and α ∈ A define the sets∆ θ t ( α, ε ) := { X ∈ R m × n : | g t M θ X α | < ε } (42)and ∆ θ t ( ε ) := [ α ∈A ∆ θ t ( α, ε ) = { X ∈ R m × n : inf α ∈A | g t M θX α | < ε } . For η >
0, define the function ψ η : T R + : t ψ η t := 2 − ησ ( t ) (43)where σ ( t ) := t + · · · + t m + n , and consider the lim sup set given byΛ θ T ( ψ η ) := lim sup t ∈ T ∆ θ t ( ψ η t ) . (44)In the case θ = , we write Λ T ( ψ η ) for Λ θ T ( ψ η ). The following result provides a reformu-lation of the set A θ m,n in terms of the lim sup sets given by (44).17 roposition 5 There exists a countable subset T of R m + n satisfying ( ) such that X t ∈ T − η σ ( t ) < ∞ ∀ η > and A θ m,n = [ η> Λ θ T ( ψ η ) ∀ θ ∈ R m . (46)Now, let µ be a measure on R m × n that is strongly contracting. Then, as a consequence of(38) and Proposition 5, the proof of Theorem 1B is reduced to showing that µ (Λ T ( ψ η )) = 0 ∀ η > ⇒ µ (Λ θ T ( ψ η )) = 0 ∀ η > T satisfying (39) and (45). It is worth mentioning, that in the proofof Proposition 5 an explicit choice of T is given. Remark.
In the case of Theorem 1A, the analogous reformulation and reduction areequally valid. The only difference is that A θ m,n is defined in terms of w ( X , θ ) rather than w × ( X , θ ). Given s = ( s , . . . , s m ) ∈ Z m + and l = ( l , . . . , l n ) ∈ Z n + , let σ ( s ) := m X j =1 s j , σ ( l ) := n X i =1 l i and ζ := ζ ( s , l ) = σ ( s ) − σ ( l ) m + n , where Z + is the set of non-negative integers. Furthermore, define the ( m + n )-tuple t =( t , . . . , t m + n ) by setting t := (cid:16) s − ζ , . . . , s m − ζ , l + ζ , . . . , l n + ζ (cid:17) (48)and let T := { t ∈ R m + n defined by (48) : s ∈ Z m + , l ∈ Z n + with σ ( s ) > σ ( t ) } . (49)The goal is to show that this choice of T is suitable within the context of Proposition 5.Equality (39) readily follows from (48). Next, it is easily verified that for any t ∈ T σ ( t ) = σ ( s ) − mζ = σ ( l ) + nζ , (50)18here σ ( t ) := P m + nk =1 t k . This together with the fact that ζ is non-negative, yields that σ ( l ) σ ( t ) σ ( s ) . (51)Furthermore, on summing the two different expressions for σ ( t ) arising in (50) and usingthe fact that σ ( l ) >
0, we obtain that σ ( t ) = σ ( s ) + σ ( l ) − m − nm + n (cid:0) σ ( s ) − σ ( l ) (cid:1) > σ ( s ) + σ ( l ) − | m − n | m + n (cid:0) σ ( s ) + σ ( l ) (cid:1) > m + n (cid:0) σ ( s ) + σ ( l ) (cid:1) . (52)The latter inequality establishes (45). In turn, it follows that for any v ∈ R + { t ∈ T : σ ( t ) < v } < ∞ . (53)Now to establish the set equality (46), fix θ ∈ R m . It is easily verified that X ∈ A θ m,n if and only if there exists an ε >
0, such that for arbitrarily large
Q > α = ( p , q ) ∈ A := Z m × ( Z n r { } ) satisfying | X q + p + θ | / Q(cid:0) X q + p + θ (cid:1) < Q − (1+ ε ) and Q + ( q ) Q . (54)
Step 1.
We show that A θ m,n ⊆ [ η> Λ θ T ( ψ η ) . (55)Suppose X ∈ A θ m,n . It follows that (54) is satisfied for infinitely many Q ∈ Z + . For anysuch Q , there exist unique s ∈ Z m + and l ∈ Z n + such that2 − s j max n | X j q + p j + θ j | , Q − (1+ ε ) o < − s j +1 for 1 j m (56)and 2 l i max { , | q i |} < l i +1 for 1 i n . (57)Throughout, X j := ( x j, , . . . , x j,n ) denotes the j -th row of X ∈ R m × n . By (56) and (57),we have that2 σ ( l ) Q + ( q ) and 2 − σ ( s ) < max nQ(cid:0) X q + p + θ (cid:1) , Q − (1+ ε ) o . This together with (54) implies that 2 − σ ( s ) < − σ ( l )(1+ ε ) . Hence, σ ( s ) − σ ( l ) > εσ ( l ) > . (58)Thus, t given by (48) with s = ( s , . . . , s m ) and l = ( l , . . . , l n ) satisfying (56) and (57) isin T . 19f σ ( s ) σ ( l ), then ζ = σ ( s ) − σ ( l ) m + n ( ) > εσ ( l ) m + n > εσ ( s )2( m + n ) ( ) > εσ ( t )4( m + n ) . If σ ( s ) > σ ( l ), then ζ = σ ( s ) − σ ( l ) m + n > σ ( s )2( m + n ) ( ) > σ ( t )4( m + n ) . On combining the above inequalities, we deduce that ζ > η σ ( t ) with η := min { ε, } m + n ) . (59)With reference to (40), we have that g t = 2 − ζ diag { s , . . . , s m , − l , . . . , − l n } and in view of (56) and (57), it follows thatinf α ∈A | g t M θX α | < · − ζ . (60)For 0 < η < η , (59) together with (60) implies thatinf α ∈A | g t M θX α | < − η σ ( t ) (61)for all sufficiently large σ ( t ). Note that (54) and (56) ensure that σ ( s ) → ∞ as Q → ∞ .Therefore, in view of (52) and the fact that (54) is satisfied for infinitely many Q ∈ Z + ,we have that (61) is satisfied for infinitely many t ∈ T . The upshot is that X ∈ Λ θ T ( ψ η )for any η ∈ (0 , η ). This establishes (55). Step 2.
We show that A θ m,n ⊇ [ η> Λ θ T ( ψ η ) . (62)Suppose X ∈ Λ θ T ( ψ η ) for some η >
0. By definition, (61) is satisfied for infinitely many t ∈ T . For any such t , there exists α = ( p , q ) ∈ A such that | g t M θX α | < − ησ ( t ) . On taking the product of the first m coordinates of g t M θX α , we obtain that m Y j =1 t j | X j q + p j + θ j | < − mη σ ( t ) . n non-zero coordinates of g t M θX α yields that Y i nq i =0 − t m + i | q i | < − n η σ ( t ) . By definition, t m + i > i n ) for any t = ( t , . . . , t m + n ) ∈ T . Also, in view of (52)we have that σ ( t ) >
0. Hence, by (39) the above displayed inequalities imply that
Q(cid:0) X q + p + θ (cid:1) < − mη σ ( t ) − σ ( t ) / and Q + ( q ) < σ ( t ) / . (63)By (53), we have that (63) is satisfied for arbitrarily large σ ( t ). Now set Q = 2 σ ( t ) / and ε := 2 mη . It follows that (54) is satisfied for arbitrarily large Q . The upshot is that X ∈ A θ m,n . This establishes (62).Steps 1 and 2 establish (46) and complete the proof of Proposition 5. ⊠ Remark.
With A θ m,n := { X ∈ R m × n : w ( X , θ ) > } , as is the case when dealingwith Theorem 1A, the proof of Proposition 5 remains pretty much unchanged. The maindifference is the manner in which we define the set T . Given s ∈ Z + and l ∈ Z + , let s := ( s, . . . , s ) ∈ Z m + and l := ( l . . . , l ) ∈ Z n + . On keeping the same notation as above, wehave that ζ = ms − nlm + n . Furthermore, define t = ( t , . . . , t m + n ) by setting t := (cid:0) s − ζ , . . . , s − ζ | {z } m times , l + ζ , . . . , l + ζ | {z } n times (cid:1) (48 ′ )and let T := { t ∈ R m + n defined by (48 ′ ) : s ∈ Z + , l ∈ Z + with s > t } . (49 ′ )Note that T is a subset of the set defined by (49). Thus, conditions (39) and (45) areautomatically satisfied for this ‘smaller’ choice of T . To establish the set equality (46),we start by verifying that X ∈ A θ m,n if and only if there exists an ε >
0, such thatfor arbitrarily large
Q > α = ( p , q ) ∈ A = Z m × ( Z n r { } ) satisfying k X q + p + θ k m < Q − − ε and | q | n Q . These inequalities replace those appearing in (54)and by naturally modifying the arguments setout in Steps 1 and 2 above, we obtain (46). In this section we develop a general framework that allows us to transfer zero measurestatements for homogeneous lim sup sets to inhomogeneous lim sup sets. To a certainextent, the framework is motivated by our desire to establish the specific transferencegiven by (47) and thereby complete the proof of Theorem 1.21et (Ω , d ) be a locally compact metric space. Given two countable ‘indexing’ sets A and T , let H and I be two maps from T × A × R + into the set of open subsets of Ω suchthat H : ( t , α, ε ) ∈ T × A × R + H t ( α, ε )and I : ( t , α, ε ) ∈ T × A × R + I t ( α, ε ) . Furthermore, let H t ( ε ) := [ α ∈A H t ( α, ε ) and I t ( ε ) := [ α ∈A I t ( α, ε ) . (64)Next, let Ψ denote a set of functions ψ : T → R + : t ψ t . For ψ ∈ Ψ , consider thelim sup sets Λ H ( ψ ) = lim sup t ∈ T H t ( ψ t ) and Λ I ( ψ ) = lim sup t ∈ T I t ( ψ t ) . (65)For reasons that will soon become apparent, we refer to sets associated with the map Has homogeneous sets and those associated with the map I as inhomogeneous sets. Thefollowing ‘intersection’ property states that the intersection of two distinct inhomogeneoussets is contained in a homogeneous set. The intersection property.
The triple (H , I , Ψ ) is said to satisfy the intersectionproperty if for any ψ ∈ Ψ , there exists ψ ∗ ∈ Ψ such that for all but finitely many t ∈ T and all distinct α and α ′ in A we have thatI t ( α, ψ t ) ∩ I t ( α ′ , ψ t ) ⊂ H t ( ψ ∗ t ) . (66)The following notion of ‘contracting’ is the natural generalisation of the R m × n versionstated in § The contracting property.
Let µ be a non-atomic, finite, doubling measure supportedon a bounded subset S of Ω. We say that µ is contracting with respect to ( I , Ψ ) if for any ψ ∈ Ψ there exists ψ + ∈ Ψ and a sequence of positive numbers { k t } t ∈ T satisfying X t ∈ T k t < ∞ , (67)such that for all but finitely t ∈ T and all α ∈ A there exists a collection C t ,α of balls B centred at S satisfying the following conditions : S ∩ I t ( α, ψ t ) ⊂ [ B ∈C t ,α B (68)22 ∩ [ B ∈C t ,α B ⊂ I t ( α, ψ + t ) (69)and µ (cid:16) B ∩ I t ( α, ψ t ) (cid:17) k t µ (5 B ) . (70)The intersection and contracting properties enable us to transfer zero µ -measure state-ments for the homogeneous lim sup sets Λ H ( ψ ) to the inhomogeneous lim sup sets Λ I ( ψ ). Theorem 5 (Inhomogeneous Transference Principle)
Suppose that (H , I , Ψ ) satis-fies the intersection property and that µ is contracting with respect to (I , Ψ ) . Then µ (Λ H ( ψ )) = 0 ∀ ψ ∈ Ψ = ⇒ µ (Λ I ( ψ )) = 0 ∀ ψ ∈ Ψ . (71)Before proving Theorem 5, we consider an application that establishes (47) and at thesame time clarifies the above abstract setup. Given the Inhomogeneous Transference Principle, we are fully armed to complete the proofof Theorem 1. In view of the reformulation and reduction carried out in §
4, both parts ofTheorem 1 follow on establishing (47) with an appropriate choice of T . As in §
4, we willconcentrate on the proof of Theorem 1B – part (B) of Theorem 1.Throughout θ ∈ R m is fixed. Let µ be a measure on R m × n that is strongly contractingalmost everywhere and fix a set T arising from Proposition 5. In terms of establishing (47),sets of µ -measure zero are irrelevant. Therefore we can simply assume that µ is stronglycontracting. We show that (47) falls within the scope of the above general framework. LetΩ := R m × n and let A be given by (41). Given ε ∈ R + , t ∈ T and α ∈ A letH t ( α, ε ) := ∆ t ( α, ε ) = ∆ ( α, ε ) and I t ( α, ε ) := ∆ θ t ( α, ε ) , where ∆ θ t ( α, ε ) is defined by (42). This defines the maps H and I associated with thegeneral framework. It is readily seen that H t ( ε ) = ∆ ( ε ) and I t ( ε ) = ∆ θ t ( ε ). Next, let Ψ be the class of functions given by (43). Then, it immediately follows thatΛ H ( ψ ) = Λ T ( ψ ) := Λ ( ψ ) and Λ I ( ψ ) = Λ θ T ( ψ ) , where the set Λ θ T ( ψ ) is defined by (44). The upshot is that (47) and (71) are precisely thesame statement. In view of the Inhomogeneous Transference Principle, it follows that (47)is a consequence of verifying that (H , I , Ψ ) satisfies the intersection property and that µ iscontracting with respect to (I , Ψ ). 23 erifying the intersection property. Let ψ ∈ Ψ . This means that ψ t = 2 − η σ ( t ) for someconstant η >
0. To establish the intersection property given by (66), define ψ ∗ by setting ψ ∗ t = 2 − η/ σ ( t ) . Obviously, ψ ∗ ∈ Ψ . Next, fix two distinct α and α ′ in A . By definition, α = ( p , q ) and α ′ = ( p ′ , q ′ )for some p , p ′ ∈ Z m and q , q ′ ∈ Z n r { } . Now, take any point X ∈ I t ( α, ψ t ) ∩ I t ( α ′ , ψ t ) := ∆ θ t ( α, ψ t ) ∩ ∆ θ t ( α ′ , ψ t ) . Clearly, we may as well assume that the intersection is non-empty. In the notation of § | g t M θ X α | < ψ t and | g t M θ X α ′ | < ψ t . (72)Let α ′′ := ( p ′′ , q ′′ ), where p ′′ := p − p ′ ∈ Z m and q ′′ := q − q ′ ∈ Z n . Since T satisfies (45)and therefore (53), it follows that | g t M X α ′′ | = | g t M X ( α − α ′ ) | = | g t M θ X α − g t M θ X α ′ | ( ) < ψ t < ψ ∗ t (73)for all but finitely many t ∈ T . If q ′′ = , we obtain via (73) that | p ′′ | < ψ ∗ t < t ∈ T . However p ′′ ∈ Z m and so we must have p ′′ = . This contradicts theassumption that α = α ′ . The upshot is that q ′′ = and so α ′′ ∈ A . Hence, it follows that X ∈ ∆ t ( α ′′ , ψ ∗ t ) ⊂ ∆ t ( ψ ∗ t ) := H t ( ψ ∗ t )for all but finitely many t ∈ T . This verifies the intersection property. Verifying the contracting property.
Recall, µ is a measure on R m × n that is strongly con-tracting. Therefore, µ is by definition non-atomic, doubling and finite. Also without lossof generality we can assume that the support S of µ is bounded. Thus, to establish that µ is contracting with respect to (I , Ψ ) it remains to verify the conditions given by (67)– (70). Fix ψ ∈ Ψ . Then, ψ t = 2 − η σ ( t ) for some constant η > ψ + bysetting ψ + t := √ ψ t . Obviously, ψ + ∈ Ψ . Let r be the positive constant appearing in thedefinition of strongly contracting. Since T satisfies (45) and therefore (53), it follows that ψ + t min { , r } and σ ( t ) > t ∈ T . Now fix such a t = ( t , . . . , t m + n ) ∈ T and α ′ = ( p , q ) ∈ A .The set I t ( α ′ , ψ t ) corresponds to X ∈ R m × n satisfying | X j q + p j + θ j | < − t j ψ t (1 j m ) with | q i | < t m + i ψ t (1 i n ) . (75)Similarly, I t ( α ′ , ψ + t ) corresponds to X ∈ R m × n satisfying | X j q + p j + θ j | < − t j ψ + t (1 j m ) with | q i | < t m + i ψ + t (1 i n ) . (76)24ithout loss of generality, we assume that the right hand side inequalities of (75) and (76)are fulfilled for α ′ = ( p , q ). Otherwise, the sets under consideration are empty and theconditions (68) – (70) are easily met. For j ∈ { , . . . , m } , define ε j = ε j, t := 2 − t j ψ + t / | q | and let δ = δ t := ψ + t . (77)By (39) and the fact that σ ( t ) >
0, it follows that P mj =1 t j > j ∈ { , . . . , m } such that 2 − t j
1. Since q ∈ Z n r { } we have that | q | −
1. Thereforemin j m ε j, t < ψ + t and (74) implies thatmin j m ε j, t < r and δ t < . By the definition of ψ + t , we have that δε j = 2 − t j ψ t / | q | for each j ∈ { , . . . , m } . Therefore,by (75) and (76), it follows thatI t ( α ′ , ψ t ) = L ( δ ε ) a , b and I t ( α ′ , ψ + t ) = L ( ε ) a , b , (78)where L ( δ ε ) a , b and L ( ε ) a , b are defined by (12) with a := q / | q | and b := ( p + θ ) / | q | . By thedefinition of strongly contracting, for each X ∈ L ( δ ε ) a , b ∩ S there is an open ball B = B X centred at X such that (13) and (14) are satisfied. With reference to the general framework,define C t ,α ′ to be the collection of all such balls. By definition, each point X ∈ L ( δ ε ) a , b ∩ S is the centre of a ball in C t ,α ′ . Finally, for any t ∈ T , let k t := C ( ψ + t ) α where C and α are the constants appearing in the definition of strongly contracting. Then(67) follows from (45);(68) follows from the definition of C t ,α ′ ;(69) and (70) follow from (13) and (14) via (78).This verifies that µ is contracting with respect to (I , Ψ ). ⊠ Remark.
When dealing with Theorem 1A, the above arguments remain essentially un-changed. The main difference is that we work with the specific T given by (49 ′ ). Thenfor any t = ( t , . . . , t m + n ) ∈ T , we have that t = · · · = t m . This ensure that ε , . . . , ε m as defined by (77) are all equal and therefore µ being contracting rather than stronglycontracting is sufficient. 25 Preliminaries for Theorem 5
In this section we group together various self contained statements that we appeal to duringthe course of establishing the Inhomogeneous Transference Principle. We start with a basiccovering result from geometric measure theory.
Lemma 2
Every collection C of balls of uniformly bounded diameter in a metric spacecontains a disjoint subcollection G such that [ B ∈C B ⊂ [ B ∈G B .
This covering lemma is usually referred to as the 5 r -lemma. For further details andproof the reader is refereed to [24, 32]. The following enables us to bound the cardinalityof the disjoint collection arising from the 5 r -lemma. Lemma 3
Let (Ω , d ) be a metric space equipped with a finite measure µ . Then every dis-joint collection C of µ -measurable subsets of Ω with positive µ -measure is at most countable.Proof. For k ∈ Z , let C ( k ) be the subcollection of C consisting of sets B ∈ C such that2 k µ ( B ) < k +1 . Obviously we have that C = S k ∈ Z C ( k ) . Since the balls in C are pairwisedisjoint, it follows that C ( k ) − k µ (Ω) < ∞ . Therefore, C is a countable union of finitesets and so is at most countable. ⊠ The next statement is a simple consequence of the continuity of measures.
Lemma 4
Let (Ω , d ) be a metric space equipped with a measure µ . Let { A i } i ∈ N be asequence of µ -measurable subsets of Ω and A ∞ := lim sup i →∞ A i = ∞ \ m =1 ∞ [ i = m A i . Then µ ( A ∞ ) = 0 if and only if for any ε > , there exists a positive constant m ( ε ) suchthat µ ( S ∞ i = m A i ) < ε for all m > m ( ε ) . Recall, that the Borel-Cantelli lemma from probability theory states that µ ( A ∞ ) = 0 if ∞ X i =1 µ ( A i ) < ∞ . Sprindzuk’s proof of Mahler’s conjecture is based on the notions of essential and inessen-tial domains – see [38, § µ -essential and µ -inessential balls. The following lemma enables us toexploit these key notions. 26 emma 5 Let (Ω , d ) be a metric space equipped with a finite measure µ . Let { B i } i ∈ N bea sequence of µ -measurable subsets of Ω . Suppose there exists a constant c ∈ (0 , suchthat for every i ∈ N µ (cid:16) B i ∩ S j = i B j (cid:17) c µ ( B i ) . (79) Then ∞ X i =1 µ ( B i ) − c µ (Ω) . Proof.
Given i ∈ N , let B i := B \ [ j = i B j and B i := B ∩ [ j = i B j . Thus, B i corresponds to the region of B i that is disjoint from the sets B j with j = i .On the other hand, B i corresponds to the region of B i that is non-disjoint from thesets B j with j = i . Obviously B i and B i are µ -measurable. By (79), we have that µ ( B i ) = µ ( B ) − µ ( B i ) > (1 − c ) µ ( B ). Thus, µ ( B ) − c µ ( B i ) . (80)By construction, B i ∩ B j = ∅ for distinct i, j ∈ N . Therefore ∞ X i =1 µ ( B i ) ( ) − c ∞ X i =1 µ ( B i ) = 11 − c µ (cid:16) ◦ [ i ∈ N B i (cid:17) − c µ (Ω) . ⊠ Fix any ψ ∈ Ψ. The goal is to show that µ (Λ I ( ψ )) = 0 . (81)We are given that µ is contracting with respect to (I , Ψ ). Note that we can assume thatthis contacting property and indeed the intersection property defined in §
5, are valid forall t ∈ T rather than all but finitely many – removing a finite number of elements from T does not alter the lim sup set Λ I ( ψ ) under consideration. With this in mind, let ψ + ∈ Ψ be the function, { k t } be the sequence and C t ,α be the collection of balls arising from thecontracting property. Since S = supp µ is bounded, we can assume that the balls in C t ,α areof radius bounded by r = diam( S ). Indeed, if C t ,α contains a ball B of radius bigger than27 centered at x then we can replace B with B ( x, r ). This replacement would not affectthe properties (68)–(70) as S ⊂ B ( x, r ). Thus, in view of Lemma 2 there is a disjointsubcollection G t ,α of C t ,α such that [ B ∈C t ,α B ⊂ [ B ∈G t ,α B . (82)Every ball B ∈ G t ,α is centred at the support of µ and so is of positive µ -measure. Thus,by Lemma 3, the collection G t ,α is at most countable. In view of (68) and (82) we havethat S ∩ I t ( α, ψ t ) ⊂ [ B ∈G t ,α B ∩ I t ( α, ψ t ) ⊂ I t ( α, ψ t ) . Taking the union over all α ∈ A and using (64) gives S ∩ I t ( ψ t ) ⊂ [ α ∈A [ B ∈G t ,α B ∩ I t ( α, ψ t ) ⊂ I t ( ψ t ) . In view of (65), it follows that S ∩ Λ I ( ψ ) ⊂ lim sup t ∈ T [ α ∈A [ B ∈G t ,α B ∩ I t ( α, ψ t ) ⊂ Λ I ( ψ ) . This implies that µ (Λ I ( ψ )) = µ (cid:16) lim sup t ∈ T [ α ∈A [ B ∈G t ,α B ∩ I t ( α, ψ t ) (cid:17) . (83)Thus, (81) will follow on establishing that the right hand side of (83) is zero. The keythat enables us to do precisely this, is the following decomposition of the right hand sidelim sup set in terms of µ -essential and µ -inessential balls. Definition.
A ball B ∈ G t ,α is said to be µ -essential if µ (cid:16) B ∩ [ α ′ ∈A r { α } [ B ′ ∈G t ,α ′ B ′ (cid:17) µ ( B ) (84)and µ -inessential otherwise.Let D t ,α denote the collection of µ -essential balls in G t ,α and let N t ,α denote the collectionof µ -inessential balls in G t ,α . Consider the corresponding limsup setsΛ D := lim sup t ∈ T [ α ∈A [ B ∈D t ,α B ∩ I t ( α, ψ t ) , and Λ N = lim sup t ∈ T [ α ∈A [ B ∈N t ,α B ∩ I t ( α, ψ t ) .
28t is easily seen that the limsup set in the right hand side of (83) is equal to Λ N ∪ Λ D andso we have that µ (Λ I ( ψ )) = µ (Λ D ∪ Λ N ) . Thus, the statement of Theorem 5 will follow on showing that µ (Λ D ) = 0 and µ (Λ N ) = 0. µ -essential case: µ (Λ D ) = 0 In view of (84), Lemma 5 and the fact that G t ,α is a disjoint collection of balls, we havethat X α ∈A X B ∈D t ,α µ ( B ) µ (Ω) . (85)It follows that µ (cid:16) [ α ∈A [ B ∈D t ,α B ∩ I t ( α, ψ t ) (cid:17) X α ∈A X B ∈D t ,α µ (cid:16) B ∩ I t ( α, ψ t ) (cid:17) ( ) X α ∈A X B ∈D t ,α k t µ (5 B ) ( ) k t λ X α ∈A X B ∈D t ,α µ ( B ) ( ) µ (Ω) λ k t . Since µ (Ω) < ∞ , the quantity 2 µ (Ω) λ is a finite positive constant. Hence, X t ∈ T µ (cid:16) [ α ∈A [ B ∈D t ,α B ∩ I t ( α, ψ t ) (cid:17) ≪ X t ∈ T k t ( ) < ∞ . A simple consequence of the Borel-Cantelli lemma is that µ (Λ D ) = 0 . µ -inessential case: µ (Λ N ) = 0 For a ball B ∈ N t ,α , let B N := B ∩ [ α ′ ∈A r { α } [ B ′ ∈G t ,α ′ B ′ . By definition, µ ( B N ) > µ ( B ) . (86)29onsider the following two setsΛ ′ N = lim sup t ∈ T [ α ∈A [ B ∈N t ,α B and Λ ′′ N = lim sup t ∈ T [ α ∈A [ B ∈N t ,α B N . Obviously we have that Λ N ⊂ Λ ′ N . Hence, it suffices to show that µ (Λ ′ N ) = 0 . Step 1: µ (Λ ′′ N ) = 0 . Note that in view of (69), for any B ∈ N t ,α ⊂ G t ,α we have that S ∩ B ⊂ I( α, ψ + t ) . (87)By definition, if x ∈ S ∩ B N then there exists α ′ ∈ A r { α } and a ball B ′ ∈ G t ,α ′ such that x ∈ B ′ . Again, by (69) we have that S ∩ B ′ ⊂ I( α ′ , ψ + t ) . (88)Since x ∈ S ∩ B ∩ B ′ , it follows via (87) and (88) that x ∈ I( α, ψ + t ) ∩ I( α ′ , ψ + t ) . We are given that (H , I , Ψ ) satisfies the intersection property. Thus, in view of (66) weconclude that x ∈ H t ( ψ ∗ t ) , where ψ ∗ ∈ Ψ is associated with ψ + . The upshot is that if x ∈ S ∩ Λ ′′ N , then x liesin the ‘homogeneous’ sets H t ( ψ ∗ t ) for infinitely many t ∈ T . In other words, S ∩ Λ ′′ N ⊂ Λ H ( ψ ∗ ). However, homogeneous lim sup sets are assumed to be of µ -measure zero - see(71). Therefore, µ (Λ ′′ N ) = 0 . (89) Step 2: µ (Λ ′ N ) = 0 . Fix any enumeration { t l } l ∈ N of the set T . For m ∈ N , letΛ ′′ N ( m ) := [ l > m [ α ∈A [ B ∈N t l,α B N . (90)Then, Λ ′′ N = T ∞ m =1 Λ ′′ N ( m ). In view of (89), it follows via Lemma 4 that for any δ > m ( δ ) such that µ (cid:0) Λ ′′ N ( m ) (cid:1) < δ ∀ m > m ( δ ) . (91)Let G ( m ) be the collection of balls 5 B such that B ∈ N t l ,α with l > m and α ∈ A . By(90), for any 5 B ∈ G ( m ) we have that B ∩ Λ ′′ N ( m ) ⊃ B N . Therefore, µ ( B ∩ Λ ′′ N ( m )) > µ ( B N ) ( ) > µ ( B ) (92)30or any 5 B ∈ G ( m ). Recall that S is bounded and so G ( m ) is a collection of balls ofuniformly bounded diameter. Thus, in view of Lemma 2 there is a disjoint subcollection G ′ ( m ) of G ( m ) such that [ B ∈G ( m ) B ⊂ [ B ∈G ′ ( m ) B .
Finally, for m > m ( δ ) let Λ ′ N ( m ) := [ l > m [ α ∈A [ B ∈N t l,α B .
Then µ (Λ ′ N ( m )) µ (cid:16) [ B ∈G ′ ( m ) B (cid:17) X B ∈G ′ ( m ) µ (cid:0) B (cid:1) ( ) λ X B ∈G ′ ( m ) µ (cid:0) B (cid:1) ( ) λ X B ∈G ′ ( m ) µ (cid:0) B ∩ Λ ′′ N ( m ) (cid:1) = 2 λ µ (cid:16) ◦ [ B ∈G ′ ( m ) B ∩ Λ ′′ N ( m ) (cid:17) λ µ (cid:0) Λ ′′ N ( m ) (cid:1) ( ) λ δ . Since Λ ′ N = T ∞ m =1 Λ ′ N ( m ), it follows via Lemma 4 that µ (Λ ′ N ) = 0. ⊠ In principle, there are numerous problems that can be treated by applying the basic recipesintroduced in this paper – in particular the Inhomogeneous Transference Principle of § Beyond simultaneous and dual extremality: d -extremality. The dual theory ofDiophantine approximation is concerned with approximation of points in R n by ( n − R n by 0–dimensionalrational planes; i.e. rational points. As a consequence of Khintchine’s transference prin-ciple, the two forms of approximation lead to equivalent notions of extremality in thehomogeneous case. However, as a consequence of a recent work by Laurent a lot more istrue. For d ∈ { , . . . n − } , it is natural to consider the Diophantine approximation theoryin which points in R n are approximated by d -dimensional rational planes – the dual andsimultaneous theories just represent the extreme. The related homogeneous Diophantineexponents ω d ( x ) have been studied in some depth by Schmidt [35] in the sixties and morerecently by Laurent [30]. We refer the reader to Laurent [30] for the definition of these ex-ponents. For the purpose of this discussion, it suffices to say that for any d ∈ { , . . . n − } ,we have that ω d ( x ) > ( d + 1) / ( n − d ) for all x ∈ R n . Now let µ be a measure on R n andsay that µ is d –extremal if ω d ( x ) = ( d + 1) / ( n − d ) for µ -almost all x ∈ R n .Khintchine’s transference principle implies that the extreme cases ( d = 0 and d = n − d –extremal are equivalent. Leaving aside the details, the Schmidt-Laurent ‘Going-up’and ‘Going-down’ transference inequalities imply that all n notions of d –extremality are infact equivalent. This together with Theorem KLW implies the following statement. Theorem 6
Any friendly measure on R n is d -extremal for all d ∈ { , . . . n − } . Given the equivalence of the n notions of d –extremality, there is no need to distinguishbetween them. Indeed, for measures µ on R n it makes perfect sense to redefine the standardnotion of extremal and say that µ is extremal if µ is d –extremal for all d ∈ { , . . . n − } .Recall, that the standard notion only requires that µ is d –extremal for d = 0 and d = n − d -extremality to the inhomogeneoussetup. Here an approximating d -dimensional rational plane is shifted by an appropriatelyscaled transversal vector parameterised by θ ∈ R n − d – the inhomogeneous part. Leavingaside the details, let ω d ( x , θ ) denote the inhomogeneous Diophantine exponent that arisesin this way. Then, given a measure µ on R n we say that µ is inhomogeneously d –extremal if for all θ ∈ R n − d ω d ( x , θ ) = ( d + 1) / ( n − d ) for µ -almost all x ∈ R n .In view of the above case for redefining the (homogeneous) notion of extremal for measures µ on R n , it would be quite natural to say that µ is inhomogeneously extremal if µ isinhomogeneously d –extremal for all d ∈ { , . . . n − } . The definition given in § µ is inhomogeneously d –extremal for d = 0 and d = n −
1. In any case, theproblem of establishing the inhomogeneous generalisation of Theorem 6 now arises.
Conjecture 1
Any friendly measure on R n is inhomogeneously d -extremal for all d ∈{ , . . . n − } . n notions of inhomogeneously d –extremality are not necessarily equivalent.Consequently, each one needs to be considered separately. Clearly, Theorem 2 implies thedesired statement for d = 0 and d = n − d between these extreme values as in the homogeneous case. For measures µ on R n one candefine the notion of d -contracting by replacing the planes L a , b appearing in the definitionof contracting (Definition 3) with d -dimensional planes. Conjecture 1 would then follow onestablishing the following generalization of Theorem 1A subject to showing that friendlymeasures are d -contracting. Conjecture 2
Let µ be a measure on R n and d ∈ { , . . . n − } . If µ is d -contractingalmost everywhere then µ is d -extremal ⇐⇒ µ is inhomogeneously d -extremal. In order to establish Conjecture 2, it would be natural to follow the recipe used to establishTheorem 1A. This involves obtaining upper and lower bounds for ω d ( x , θ ) separately. Inshort, the upper bound would follow on applying the Inhomogeneous Transference Prin-ciple. Of course, this is subject to being able to reformulate the upper bound problemanalogues to that of § §
5. The lower boundwould follow on establishing an analogue of Theorem BL or at least an analogue of Cas-sels’ Theorem VI in [20, Chapter 5] within the setting of approximating points in R n by d -dimensional planes. More precisly, concerning the former we seek a lower bound for ω d ( x , θ ) in terms of the uniform homogeneous exponent b ω d ′ ( x ) for some 0 ≤ d ′ ≤ n − Conjecture 3
Let x ∈ R n and d ∈ { , . . . , n − } . Then for all θ ∈ R n − d ω d ( x , θ ) > b ω n − − d ( x ) . Note that Conjecture 3 coincides with the left hand side inequality of (37) in the extremecases d = 0 and d = n − Beyond extremality in R n : system of linear forms. The theory of inhomogeneousextremality for measures on R m × n corresponding to a genuine system of linear forms isnot covered by Theorem 2 or indeed Conjecture 1 above. By genuine we simply meanthat both m and n are strictly greater than one and so we are outside of the dual andsimultaneous theories. Naturally it would be highly desirable to establish a general resultfor measures on R m × n analogues to Theorem 2 or simply Theorem 3. However, such a resultis currently non-existent even in the homogeneous setting for manifolds and constitutes a33ey open problem. Indeed, the homogeneous ‘manifold’ problem first eluded to in [29, § Beyond extremality: Khintchine-Groshev type results.
The inhomogeneous ex-tremality results obtained in this paper constitute the first step towards developing a co-herent inhomogeneous theory for manifolds in line with the homogeneous theory [4, 7, 16].It would be desirable to adapt the techniques of this paper to obtain the inhomogeneousanalogues of the convergence Khintchine-Groshev type results.
Beyond Euclidean spaces : C , Q p and S -arithmetic. In a nutshell, Theorem 1 enablesus to transfer homogeneous extremality statements for measures on a Euclidean space toinhomogeneous statements. It would be desirable to obtain analogous results within the p -adic, complex or more generally the S -arithmetic setup. To this end, we refer the reader tothe papers [8, 9, 22, 26, 28, 33] and references within for the various homogeneous results.The Inhomogeneous Transference Principle (Theorem 5) is applicable within these non-Euclidean settings and provides a natural path for obtaining the desired inhomogeneousgeneralisations. Beyond rigid inhomogeneous approximation.
Within the context of the Diophan-tine approximation problems addressed in this paper – namely that of inhomogeneousextremality – the inhomogeneous part is arbitrary but always fixed. However, for vari-ous Diophantine approximation problems there are often major advantages in treating theinhomogeneous part as a variable. For example, in (1) we may consider the situation inwhich θ ∈ R m is a function of the approximated point X ∈ R m × n under consideration.Geometrically, on allowing θ to depend on X we perturb the underlying approximatingplanes L a , b – see (11). These ‘perturbed planes’ now play the role of the approximatingobjects and are not necessarily planes. However, the fact that the approximating objectsare no longer planes is completely irrelevant when it comes to applying the InhomogeneousTransference Principle (Theorem 5). In short, as long as the intersection and contractingproperties are satisfied the Inhomogeneous Transference Principle can be used and thenature of the approximating objects is irrelevant.To clarify the above discussion, we describe a conjecture that can be treated as aninhomogeneous problem in which the inhomogeneous part is a variable. Let ψ : N → R + be a monotonic function such that ψ ( r ) → r → ∞ and for x ∈ R consider the solubilityof the inequality | x n + a n − x n − + · · · + a x + a | < ψ (cid:0) | a | (cid:1) (93)34n integer vectors a = ( a n − , . . . , a ) ∈ Z n . Note that when the right hand side of (93) issmall, the above inequality implies that there exists an algebraic integer close to x . Thefollowing divergence statement is due to Bugeaud [18]. Theorem B
Let ψ be a monotonic function such that P ∞ q =1 q n − ψ ( q ) diverges. Then, foralmost all x ∈ R the inequality ( ) has infinitely many solutions a ∈ Z n . Establishing the convergence counterpart to Theorem B represents an open problem anal-ogous to a problem of A. Baker [2] settled by Bernik [11].
Conjecture 4
Let ψ be a monotonic function such that P ∞ q =1 q n − ψ ( q ) converges. Then,for almost all x ∈ R the inequality ( ) has at most finitely many solutions a ∈ Z n . In the special case that ψ ( q ) = q − n +1 − ε with ε >
0, the above conjecture can be viewedas the algebraic integers analogue of Mahler’s problem [31]. In terms of establishing Con-jecture 4, our approach is to view it as a problem in inhomogeneous Diophantine approx-imation in which the inhomogeneous part θ ∈ R is a variable. This is simple enoughto do! With reference to inequality (93), we let θ be x n . Thus, Conjecture 4 is equiva-lent to a ‘perturbed’ inhomogeneous Diophantine approximation problem restricted to theVeronese curve V n − in R n − . We claim that the perturbed inhomogeneous problem canbe treated via the methods introduced in this paper. We intend to return to Conjecture 4in a forthcoming paper. Acknowledgements.
SV would like to thank the dynamic duo Iona and Ayesha for pro-longing his youth – of course only in the mind! Also, happy number six girls and “oo eea a a walla walla bing bang hey!” to you too. VB would like to thank Vasili Bernik foran interesting discussion that has triggered this work. Also VB and SV wish him happy sixty one – many happy returns of the day, Basil!
Victor BeresnevichUniversity of York, Heslington, York, YO10 5DD, England
E-mail address : [email protected]
Sanju VelaniUniversity of York, Heslington, York, YO10 5DD, England
E-mail address : [email protected]
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