Twisted inhomogeneous Diophantine approximation and badly approximable sets
aa r X i v : . [ m a t h . N T ] M a y TWISTED INHOMOGENEOUS DIOPHANTINEAPPROXIMATION AND BADLY APPROXIMABLE SETS
STEPHEN HARRAP
Abstract.
For any real pair i, j ≥ i + j = 1 let Bad ( i, j ) denotethe set of ( i, j )-badly approximable pairs. That is, Bad ( i, j ) consists ofirrational vectors x := ( x , x ) ∈ R for which there exists a positiveconstant c ( x ) such thatmax n k qx k /i , k qx k /j o > c ( x ) /q ∀ q ∈ N . A new characterization of
Bad ( i, j ) in terms of ‘well-approximable’ vec-tors in the area of ‘twisted’ inhomogeneous Diophantine approximationis established. In addition, it is shown that Bad x ( i, j ), the ‘twisted’ inho-mogeneous analogue of Bad ( i, j ), has full Hausdorff dimension 2 when x is chosen from Bad ( i, j ). The main results naturally generalise the i = j = 1 / Introduction
Background – the homogeneous theory.
A classical result dueto Dirichlet states that for any real number x there exist infinitely manynatural numbers q such that(1.1) k qx k ≤ q , where k·k denotes the distance to the nearest integer. This result can easilybe generalised to higher dimensions. In particular, the following ‘weighted’simultaneous version of the above statement is valid. Choose any positivereal numbers i and j satisfying(1.2) i, j ≥ i + j = 1 . Then, for any vector x ∈ R there exist infinitely many natural numbers q such that(1.3) max n k qx k /i , k qx k /j o ≤ q . Here, without loss of generality, if i = 0 we employ the convention that k x k /i = 0 and so the above statement reduces to Dirichlet’s original result.It is natural to ask whether the right hand side of inequality (1.3) can in Mathematics Subject Classification.
Primary 11K60; Secondary 11J83.
Key words and phrases. inhomogeneous Diophantine approximation, simultaneousbadly approximable, Hausdorff dimension. general be tightened; that is, if 1 /q may be replaced by c/q for some absoluteconstant c ∈ (0 ,
1) whilst still allowing (1.3) to hold infinitely often for allreal vectors. It is still an open problem as to whether there exists an ‘optimal’constant in the sense that the statement holds only finitely often for at leastone real vector if it is replaced by any lesser constant. Conversely, in the one-dimensional setting, concerning statement (1.1), such an ‘optimal’ constant(namely 1 / √
5) was found by Hurwitz (e.g., Theorems 193 & 194 in [15,Chapter XI]).The above discussion motivates the study of real vectors x for whichthe right hand side of (1.3) cannot be improved by an arbitrary positiveconstant. Throughout, we will impose the following natural restriction onthese vectors. We say x := ( x , x ) is irrational (abbreviated irr. ) if itscomponents x i together with 1 are linearly independent over the rationals. Definition 1.1.
An irrational vector x is ( i, j ) -badly approximable if thereexists a constant c ( x ) > n k qx k /i , k qx k /j o > c ( x ) q ∀ q ∈ N . The set of all such vectors will be denoted
Bad ( i, j ).The results of this paper (for i, j >
0) do remain true when x is notassumed to be irrational in the above and later definitions. However, wechoose to avoid this degenerate case for the sake of clarity. Furthermore, allof the sets and arguments considered in this paper are invariant under inte-ger translation, so there will be no loss in generality in assuming throughoutthat all vectors are confined to the unit square (or the unit n -cube whenin higher dimensions) unless otherwise stated. Accordingly, for example, if i = 0 the set Bad (0 ,
1) will be identified with [0 , × Bad , where
Bad isthe standard one dimensional set of badly approximable numbers. In otherwords,
Bad (0 ,
1) consists of vectors x with x ∈ [0 ,
1] and x ∈ Bad := (cid:26) irr. x ∈ [0 ,
1] : ∃ c ( x ) > k qx k > c ( x ) q ∀ q ∈ N (cid:27) . Definition 1.2.
A mapping ψ : N → R is an approximating function if ψ is strictly positive and non-increasing. Definition 1.3.
For any approximating function ψ , define W ( i,j ) ( ψ ) to bethe set of vectors x ∈ [0 , such that the inequalitymax n k qx k /i , k qx k /j o ≤ ψ ( q )holds for infinitely many natural numbers q . .I.D.A. & BADLY APPROXIMABLE SETS 3 Application of the following classical theorem of Khintchine [18] yieldsthat for every pair of reals i , j satisfying (1.2) the set Bad ( i, j ) is of two-dimensional Lebesgue measure zero. Lebesgue measure will hereafter bedenoted µ . Khintchine’s Theorem (1926).
For any pair of reals i , j satisfying (1.2)and any approximating function ψ we have µ (cid:0) W ( i,j ) ( ψ ) (cid:1) = , ∞ X r =1 ψ ( r ) < ∞ . , ∞ X r =1 ψ ( r ) = ∞ . It is worth emphasising here that the choice of approximating function ψ iscompletely irrelevant once the reals i, j have been fixed. We also mentionhere that in the i = j = 1 / ψ can be relaxed (see [13] for details). However, whether this is true in generalis still an open problem.The question of whether each null set Bad ( i, j ) is non-empty was for-mally answered by Pollington & Velani [23] who showed that for everychoice of reals i , j satisfying (1.2) we have(1.4) dim ( Bad ( i, j ) ∩ Bad (1 , ∩ Bad (0 , (cid:0) [0 , (cid:1) = 2 . Here, and throughout, ‘dim’ denotes standard Hausdorff dimension. Withthis result in mind, the aim of this paper is to obtain an expression for
Bad ( i, j ) in terms of ‘well-approximable’ vectors in the area of ‘twisted’inhomogeneous Diophantine approximation.1.2. Background – the ‘twisted’ inhomogeneous theory.
Another re-sult of Khintchine (see for example [14, Chapter 10, Theorem 10 . x and any real γ there exist infinitely many naturalnumbers q such that(1.5) k qx − γ k ≤ ǫ √ q , where ǫ > γ = 0’ theorem by only the constant ǫ . When certain restrictions are placed on the choice of γ , a tighter ‘opti-mal’ inequality was found to hold by Minkowski [24]: The right hand side The arguements used by Davenport in [8] to show that
Bad (1 / , /
2) is uncountablecan easily be adapted to show that
Bad ( i, j ) is uncountable for every choice of reals i, j satisfying (1.2). S. HARRAP of (1.5) can be replaced with 1 / (4 q ) if it is assumed that γ is not of theform γ = mx + n for some integers m and n . Both of these statements leadto the implication that the sequence { qx } q ∈ N modulo one is dense in theunit interval for any irrational x . Moreover, Kronecker’s Theorem (see [20])implies that the sequence { q x } q ∈ Z modulo one is dense in [0 , for any ir-rational vector x . Furthermore, the sequence is uniformly distributed. Thisnaturally leads to the concept of approximating real vectors γ in [0 , bythe sequence { q x } q ∈ N modulo one with increasing degrees of accuracy. Forobvious reasons we call this approach ‘twisted’ Diophantine approximation. Definition 1.4.
For each fixed approximating function ψ , any irrationalvector x and each pair i , j satisfying (1.2) define W x ( i,j ) ( ψ ) to be the set ofvectors γ := ( γ , γ ) ∈ [0 , such that the inequalitymax n k qx − γ k /i , k qx − γ k /j o ≤ ψ ( | q | )holds for infinitely many non-zero integers q .Establishing a Khintchine-type law (an analogue to Khintchine’s The-orem) for the Lebesgue measure of W x ( i,j ) ( ψ ) is more difficult than in thehomogeneous case. That said, by utilising the Borel-Cantelli lemma fromprobability theory it is easy to show that for every i , j satisfying (1.2), anyirrational x and every approximating function ψ we have µ (cid:0) W x ( i,j ) ( ψ ) (cid:1) = 0 if ∞ X r =1 ψ ( r ) < ∞ . One might therefore expect that no matter what the choice of reals i , j , ir-rational x or approximating function ψ we should be able to conclude that µ (cid:0) W x ( i,j ) ( ψ ) (cid:1) = 1 if the above sum diverges. However, the following state-ment, a consequence of Theorem 6.1 (see Appendix), suggests that once thereals i , j have been fixed the set of irrational vectors for which we do obtaina set of full measure is dependent on the choice of approximating function.This subtle distinction is what makes the metrical theory in the ‘twisted’setting more delicate, and sophisticated, than its standard homogeneouscounterpart. Theorem 1.5 (Twisted Khintchine-type Theorem) . Let ψ be a fixed ap-proximating function. Then, for µ -almost all irrational vectors x ∈ [0 , µ (cid:0) W x ( i,j ) ( ψ ) (cid:1) = 1 if ∞ X r =1 ψ ( r ) = ∞ . .I.D.A. & BADLY APPROXIMABLE SETS 5 Approximating functions whose sum diverges will hereafter simply bereferred to as divergent and the set of all divergent approximating functionswill be denoted by D . Definition 1.6.
Fix a pair of reals i , j satisfying (1.2). Then, for each ψ ∈ D we define V ( i,j ) ( ψ ) := (cid:8) irr. x : µ (cid:0) W x ( i,j ) ( ψ ) (cid:1) = 1 (cid:9) . Note that Theorem 1.5 is equivalent to the statement “ µ (cid:0) V ( i,j ) ( ψ ) (cid:1) =1 for each ψ ∈ D ”. In view of this theorem we ask whether there existirrational vectors x such that a set of full measure is obtained irregardlessof the choice of divergent approximating function. In other words, we wishto characterise the set \ ψ ∈D V ( i,j ) ( ψ ) . It is certainly not obvious as to whether the intersection is non-empty.Almost all activity in the past has been centred on the specific i = j =1 / Bad (1 / , /
2) are commonly referred to as simultaneously badly approximable pairs . The most notable breakthroughwas made by Kurzweil [21], who proved the following remarkable result.
Kurzweil’s Theorem (1955). \ ψ ∈D V ( , )( ψ ) = Bad (cid:0) , (cid:1) . In fact, Kurzweil’s result was more general than the above (see § Definition 1.7.
Fix an irrational vector x ∈ [0 , and two real numbers i and j satisfying (1.2). Define Bad x ( i, j ) as the set of vectors γ ∈ [0 , forwhich there exists a constant c ( γ ) > n k qx − γ k /i , k qx − γ k /j o > c ( γ ) | q | for all q ∈ Z =0 . The set
Bad x ( i, j ) represents the twisted inhomogeneous analogue of Bad ( i, j ) introduced in § i = j = 1 / S. HARRAP result (also see the work of Tseng [30] and Moshcheivitin [25] for more recentextensions).
Theorem BHKV (2010).
For any irrational x ∈ [0 , , dim (cid:18) Bad x (cid:18) , (cid:19)(cid:19) = 2 . Once more, the statement proved was more general than the above, whichhas been simplified for our needs. At the time of writing there were no knownresults concerning the Hausdorff dimension of
Bad x ( i, j ) for a general pair i and j . 2. The main results
Statements of Results.
The following statement represents our maintheorem and generalises Kurzweil’s Theorem from the classical ‘1 / / i, j )–weightings’. Theorem 2.1.
For every pair of reals i and j satisfying (1.2), \ ψ ∈D V ( i,j ) ( ψ ) = Bad ( i, j ) . In view of Khintchine’s Theorem and statement (1.4), Theorem 2.1 im-mediately implies that the intersection on the LHS above is of 2-dimensionalLebesgue measure zero and of full Hausdorff dimension two.Our next result makes a contribution towards determining the Hausdorffdimension of
Bad x ( i, j ). Theorem 2.2.
For any real i and j satisfying (1.2) and any x ∈ Bad ( i, j ) , dim ( Bad x ( i, j )) = 2 . The proof of this theorem makes use of a general framework developedby Kristensen, Thorn & Velani [19]. This framework was designed for estab-lishing dimension results for large classes of badly approximable sets andthe above statement constitutes one further application. In all likelihoodthe above result is true without the assumption on x . Conjecture 2.3.
For any real i and j satisfying (1.2) and any irrationalvector x ∈ [0 , , dim ( Bad x ( i, j )) = 2 . It seems that the ideas of [4], which also make use of the framework in[19], are not extendable to the full weighted setting of Conjecture 2.3; anew approach may be required. Note that Theorem 2.2, together with (1.4) .I.D.A. & BADLY APPROXIMABLE SETS 7 trivially implies that the conjecture is true for a set of irrational vectors x of full dimension. Remark.
Since submission, Nikolay Moshchevitin and the named autherhave strengthened Theorem 2.2 from a statement implying full Hausdorffdimension to the statement that
Bad x ( i, j ) is ‘winning’ under the givenconditions. However, obtaining a solution to Conjecture 2.3 still remainsout of reach.2.2. Higher Dimensions.
We describe the n -dimensional generalisation ofthe sets Bad ( i, j ) and V ( i,j ) ( ψ ) along with the higher dimensional analogueof the statements in § n -tuple of reals i := i , . . . , i n ≥ P nj =1 i j = 1. We naturally define Bad ( i ) to be the set of vectors x := ( x , . . . , x n ) ∈ [0 , n for which there exists a constant c ( x ) > n k qx k /i , . . . , k qx n k /i n o > c ( x ) q ∀ q ∈ N . For any approximating function ψ and any irrational vector x ∈ [0 , n , wedenote by W xi ( ψ ) the set of vectors γ := ( γ , . . . , γ n ) ∈ [0 , n such thatmax n k qx − γ k /i , . . . , k qx n − γ n k /i n o ≤ ψ ( | q | )for infinitely many non-zero integers q . Also, set V i ( ψ ) := { x ∈ [0 , n : µ n ( W xi ( ψ )) = 1 } , where µ n denotes the standard n -dimensional Lebesgue measure, and oncemore denote by D the set of approximating functions for which ∞ X r =1 ψ ( r ) = ∞ . The proof of Theorem 2.1 can be extended in the obvious way, with no newideas or difficulties, allowing us to establish the following statement. Forevery real n -tuple i such that i , . . . , i n ≥ P nj =1 i j = 1,(2.1) \ ψ ∈D V i ( ψ ) = Bad ( i ) . Khintchine’s Theorem and statement (1.4) can also be generalised and yieldthat the above intersection is of n -dimensional Lebesgue measure zero andof full Hausdorff dimension n . As eluded to above, Kurzweil proved in [21]that equality (2.1) holds in the case that i = · · · = i n = 1 /n , for everynatural number n . This includes the one-dimensional formulation of theproblem corresponding to the set Bad . However, in these generalisations the
S. HARRAP notation gets rather awkward and so for the sake of clarity (and relevanceto the material in §
1) we will prove the ‘ n = 2’ case only.The set Bad x ( i ) can be defined in the obvious way and analogues toTheorem 2.2 and Conjecture 2.3 can easily be established. The frameworkand proof of Theorem 2.2 in § Multiplicative Diophantine Approximation
This section comprises of a brief discussion of related problems in the areaof multiplicative Diophantine approximation, where loosely speaking thesupremum norm is replaced by the geometric mean. For example, one couldconsider the set of vectors that are ‘well approximable’ in a multiplicativesense.
Definition 3.1.
Let ψ be any approximating function. Then, define W M ( ψ ) := (cid:8) x ∈ [0 , : k qx k k qx k ≤ ψ ( q ) for inf. many q ∈ N (cid:9) . The relevant measure-theoretic result concerning W M ( ψ ) was found by Gal-lagher [12] who proved a theorem implying the following. Gallagher’s Theorem (1962).
For any approximating function ψ , µ ( W M ( ψ )) = , ∞ X r =1 ψ ( r ) log(1 /ψ ( r )) < ∞ . , ∞ X r =1 ψ ( r ) log(1 /ψ ( r )) = ∞ . It is natural to develop a twisted theory for the multiplicative setup.
Definition 3.2.
Fix any approximating function ψ and any irrational vector x in [0 , . Then, define W x M ( ψ ) := (cid:8) γ ∈ [0 , : k qx − γ k k qx − γ k ≤ ψ ( | q | ) for inf. q ∈ Z =0 (cid:9) . The following statement is a consequence of Theorem 6.1 (see the Appen-dix). .I.D.A. & BADLY APPROXIMABLE SETS 9
Theorem 3.3.
Fix any approximating function ψ . Then for µ -almost allirrational vectors x ∈ [0 , we have µ ( W x M ( ψ )) = , ∞ X r =1 ψ ( r ) log(1 /ψ ( r )) < ∞ . , ∞ X r =1 ψ ( r ) log(1 /ψ ( r )) = ∞ . Once more one could ask whether there exist irrational vectors x suchthat a set of full measure is obtained irrespective of the choice of approximat-ing function. Accordingly, let D M denote the set of approximating functionsfor which P ∞ r =1 ψ ( r ) log(1 /ψ ( r )) diverges and define V M ( ψ ) := { irr. x : µ ( W x M ( ψ )) = 1 } . Consider the intersection(3.1) \ ψ ∈D M V M ( ψ ) . In view of Theorem 2.1, one might expect that (3.1) is equivalent to themultiplicative analogue of the set of badly approximable pairs. However,quite how such an analogue should be defined is up for debate.One could argue that a valid choice for a set of multiplicatively badlyapproximable numbers might be
Bad L := (cid:26) x ∈ [0 , : ∃ c ( x ) > k qx k k qx k > c ( x ) q ∀ q ∈ N (cid:27) . The famous Littlewood conjecture states that the set
Bad L is empty. Forrecent developments and background concerning the Littlewood conjecturesee [10], [22] and the references therein.Another candidate for the multiplicatively badly approximable numbersis the larger set Mad := (cid:26) x ∈ [0 , : ∃ c ( x ) > k qx k k qx k > c ( x ) q log q ∀ q ∈ N (cid:27) , recently introduced in [1]. Hence, the following question arises: Can \ ψ ∈D M V M ( ψ ) be characterized as Bad L or Mad ?Even establishing that
Bad L ⊆ T ψ ∈D M V M ( ψ ) seems non-trivial. Proof of Theorem 2.1
Proof of Theorem 2.1 (Part ). If either i = 0 or j = 0 the theoremsimplifies to a one-dimensional ‘ n = 1’ version of Kurzweil’s Theorem corre-sponding to Bad . Therefore, we can and will assume hereafter that i, j >
Proposition 4.1.
For every real i, j > such that i + j = 1 , \ ψ ∈D V ( i,j ) ( ψ ) ⊆ Bad ( i, j ) . Proof.
We will show that if x / ∈ Bad ( i, j ) then x / ∈ T ψ ∈D V ( i,j ) ( ψ ) andprove the result via a contrapositive argument. In particular, we will showthat for every such x there exists an approximating function ψ ∈ D forwhich(4.1) µ (cid:0) W x ( i,j ) ( ψ ) (cid:1) = 0;i.e., the points γ := ( γ , γ ) ∈ [0 , that satisfy the inequalitymax n k qx − γ k /i , k qx − γ k /j o ≤ ψ ( | q | )for infinitely many non-zero integers q form a null set with respect to theLebesgue measure.First, if x / ∈ Bad ( i, j ) then by definition there exists a sequence { q k } k ∈ N of non-zero integers such that(4.2) max n k q k x k /i , k q k x k /j o < c k | q k | , | q k | < | q k +1 | ∀ k ∈ N , where c k > c k → k → ∞ . Furthermore, it can be assumed that(4.3) 1 > c k > / (2 min { i,j } ) c k +1 ∀ k ∈ N . If this were not the case then we could simply choose a suitable subsequenceof { q k } . In addition, it may also be assumed that the sequence n ( c k ) − / o k ∈ N takes integer values for every index k . Note that the latter assumption, alongwith condition (4.3), guarantees that for every k (4.4) ( c k ) − ≥ . We wish to construct a divergent approximating function ψ for whichequation (4.1) is fulfilled. To that end, we introduce some useful notation.For each k ≥
1, let n k := | q k | ( c k ) − / . In view of the above assumptionsthe sequence { n k } k ∈ N is increasing and takes strictly positive integer values .I.D.A. & BADLY APPROXIMABLE SETS 11 for each index k . That said, we set n := 0 for future conciseness. Next, foreach natural number r define ψ ( r ) := ( , r ≤ n . | q k +1 | − ( c k +1 ) , n k < r ≤ n k +1 for every k ≥ . It is obvious that ψ is a decreasing and strictly positive function. To show ψ ∈ D , note that ∞ X r =1 ψ ( r ) > ∞ X k =1 n k +1 X r = n k +1 ψ ( r )= ∞ X k =1 ( n k +1 − ( n k + 1) + 1) ψ ( n k +1 )= ∞ X k =1 (cid:16) | q k +1 | ( c k +1 ) − − | q k | ( c k ) − (cid:17) | q k +1 | − ( c k +1 ) = ∞ X k =1 − | q k || q k +1 | (cid:18) c k +1 c k (cid:19) ! > ∞ X k =1 − (cid:18) c k +1 c k (cid:19) ! (since | q k | < | q k +1 | ) ( ) > ∞ X k =1 (cid:0) − − / (2 min { i,j } ) (cid:1) ≥ ∞ X k =1
12 = ∞ , as required.Finally, we endeavour to show (4.1) holds for our choice of divergentfunction. To that end, for each non-zero integer q let R ψ o ( q ) := n γ ∈ [0 , : max n k qx − γ k /i , k qx − γ k /j o ≤ ψ ( | q | ) o denote the closed rectangular region in the plane centred at the point q x (mod 1) of sidelengths 2 ψ i ( | q | ) and 2 ψ j ( | q | ) respectively. When using thenotation ‘ R ψ o ( q )’ it will be understood that i , j and x are fixed. In addition,all such closed rectangular regions will be referred to throughout as simplya ‘rectangle’ and all points within any such rectangle will tacitly be moduloone. It follows that W x ( i,j ) ( ψ ) = (cid:8) γ ∈ [0 , : γ ∈ R ψ o ( q ) for inf. many q ∈ Z =0 (cid:9) = { γ ∈ [0 , : γ ∈ n k [ | q | = n k − +1 R ψ o ( q ) for inf. many k ∈ N } . (4.5) In view of the Borel-Cantelli lemma, to show that equation (4.1) holds it isenough to show that(4.6) ∞ X k =1 µ n k [ | q | = n k − +1 R ψ o ( q ) < ∞ . We will estimate the LHS by estimating the measure of each union of rect-angles of the form R ∗ ψ o ( k ) : = n k [ | q | = n k − +1 R ψ o ( q ) , for k ∈ N . We will hereafter refer to any union of rectangles as a ‘collection’. For each k , the collection R ∗ ψ o ( k ) consists of 2( n k − n k − ) rectangles in [0 , eachcentred at some point q x for which n k − < | q | ≤ n k . By definition, everyrectangle in a collection is of the same measure, in particular each hasrespective sidelengths 2 ψ i ( n k ) and 2 ψ j ( n k ).To estimate the measure of R ∗ ψ o ( k ) we will cover it with a collectionof larger rectangles whose measure will in some sense increase at a morecontrollable rate than those of R ∗ ψ o ( k ). This will allow us to calculate afinite upper bound for the sum (4.6) as required. With these aims in mind,for each index k set S ∗ ψ o ( k ) : = | q k | [ | q | =1 ( γ ∈ [0 , : k qx − γ k ≤ n k | q k | (cid:18) c k | q k | (cid:19) i + ψ i ( n k )and k qx − γ k ≤ n k | q k | (cid:18) c k | q k | (cid:19) j + ψ j ( n k ) ) . Each collection S ∗ ψ o ( k ) now consists of 2 | q k | rectangles in [0 , , one centredat each point q x with 1 ≤ | q | < | q k | . The sidelengths of each of theserectangles are2 n k | q k | (cid:18) c k | q k | (cid:19) i + ψ i ( n k ) ! and 2 n k | q k | (cid:18) c k | q k | (cid:19) j + ψ j ( n k ) ! respectively. An upper bound for the Lebesgue measure of S ∗ ψ o ( k ) can beeasily deduced. We have(4.7) µ (cid:0) S ∗ ψ o ( k ) (cid:1) ≤ | q k | n k | q k | (cid:18) c k | q k | (cid:19) i + ψ i ( n k ) ! n k | q k | (cid:18) c k | q k | (cid:19) j + ψ j ( n k ) ! for every index k ≥ S ∗ ψ o ( k ) covers R ∗ ψ o ( k ) for each k . As the rectanglesof S ∗ ψ o ( k ) are larger than those of R ∗ ψ o ( k ), any rectangle of R ∗ ψ o ( k ) centred .I.D.A. & BADLY APPROXIMABLE SETS 13 at a point q ′ x with n k − < | q ′ | ≤ | q k | will automatically be contained in thecorresponding rectangle of S ∗ ψ o ( k ). Hence, it will suffice to check that anyrectangle of R ∗ ψ o ( k ) centred at a point q ′ x with | q k | < | q ′ | ≤ n k is coveredby some rectangle of S ∗ ψ o ( k ). It is clear by construction and inequality (4.4)that | q k | < n k and so rectangles of this type are present in every R ∗ ψ o ( k ).For each of these integers q ′ we can find a natural number m such that | q ′ − mq k | ≤ | q k | . This implies there must be a rectangles of the collection S ∗ ψ o ( k ) that is centred at the point ( q ′ − mq k ) x . It is also clear that m canalways be chosen in a way such that | mq k | < | q ′ | . It follows that(4.8) | m | < | q ′ || q k | ≤ n k | q k | . Now, consider the distance between the points q ′ x and ( q ′ − mq k ) x . We have k q ′ x − ( q ′ − mq k ) x k = k− mq k x k ≤ | m | k q k x k ( ) < | m | (cid:18) c k | q k | (cid:19) i ( ) < n k | q k | (cid:18) c k | q k | (cid:19) i , and similarly k q ′ x − ( q ′ − mq k ) x k < n k | q k | (cid:18) c k | q k | (cid:19) j . Combining the two above inequalities yields that any rectangle of R ∗ ψ o ( k )centred at a point q ′ x with | q k | < | q ′ | ≤ n k is covered by the rectangle of S ∗ ψ o ( k ) centred at ( q ′ − mq k ) x . This shows that S ∗ ψ o ( k ) is a cover for R ∗ ψ o ( k )and so ∞ X k =1 µ (cid:0) R ∗ ψ o ( k ) (cid:1) ≤ ∞ X k =1 µ (cid:0) S ∗ ψ o ( k ) (cid:1) . Estimate (4.7) yeilds that the RHS is bounded above by ∞ X k =1 | q k | n k | q k | (cid:18) c k | q k | (cid:19) i + ψ i ( n k ) ! n k | q k | (cid:18) c k | q k | (cid:19) j + ψ j ( n k ) ! = ∞ X k =1 | q k | (cid:16) ( c k ) − ( c k ) i | q k | − i + | q k | − i ( c k ) i (cid:17) × (cid:16) ( c k ) − ( c k ) j | q k | − j + | q k | − j ( c k ) j (cid:17) = 8 ∞ X k =1 | q k | | q k | − i − j (cid:16) ( c k ) i − + ( c k ) i (cid:17) (cid:16) ( c k ) j − + ( c k ) j (cid:17) . However, we have that i + j = 1 and so this reduces to8 ∞ X k =1 (cid:16) ( c k ) i + j − + ( c k ) i + j + ( c k ) i + j − + ( c k ) i + j − (cid:17) = 8 ∞ X k =1 (cid:16) c k ) + ( c k ) i + ( c k ) j (cid:17) ≤ ∞ X k =1 c k ) { i,j } / ) < ∞ X k =1 ( c ) { i,j } / − ( k − = 64 ( c ) { i,j } / < ∞ , as required. This completes the proof of Lemma 4.1. (cid:3) Proof of Theorem 2.1 (Part ). In this section we prove the com-plementary inclusion to that of Proposition 4.1.
Proposition 4.2.
For every real i, j > such that i + j = 1 , Bad ( i, j ) ⊆ \ ψ ∈D V ( i,j ) ( ψ ) . Proof.
We are required to show that if x ∈ Bad ( i, j ) then for every divergentapproximating function ψ we have that µ (cid:0) W x ( i,j ) ( ψ ) (cid:1) = 1 . To do this we first prove the intermediary result that for every x ∈ Bad ( i, j )we have(4.9) µ (cid:0) W x ( i,j ) ( ψ ) (cid:1) > ψ ∈ D .Fix x ∈ Bad ( i, j ). By definition there exists a constant c ( x ) > q max n k qx k /i , k qx k /j o > c ( x ) q . Next, choose any function ψ ∈ D . To ensure that certain technical conditionsrequired later in the proof are met we will work with a refinement of ψ . Let a ∗ := 2 − / max { i,j } and a ∗ := 2 − / min { i,j } , then for each r ∈ N set ψ ( r ) := min (cid:26) ψ ( r ) , a ∗ , a ∗ c ( x )2 | r | (cid:27) . .I.D.A. & BADLY APPROXIMABLE SETS 15 Finally, choose any integer k such that(4.10) k > r define ψ ( r ) := ( ψ ( k ) , r ≤ k.ψ ( k t +1 ) , k t < r ≤ k t +1 for each t ∈ N . It is easy to see that for each r ∈ N (4.11) ψ ( r ) ≤ ψ ( r ) ≤ ψ ( r )and that ψ ∈ D . It is also clear that ψ is decreasing and strictly positive.Furthermore, ∞ X r =1 ψ ( r ) ≥ ∞ X t =1 k t +1 X r = k t +1 ψ ( r )= ∞ X t =1 (cid:0) k t +1 − k t (cid:1) ψ ( k t +1 )= 1 k ∞ X t =1 (cid:0) k t +2 − k t +1 (cid:1) ψ ( k t +1 ) ≥ k ∞ X t =1 k t +2 X r = k t +1 +1 ψ ( r )= 1 k ∞ X r = k +1 ψ ( r ) = ∞ , and so ψ too is a divergent approximating function.With reference to § W x ( i,j ) ( ψ ) in terms of the rectangles R ψ ( q ) given by (4.5) now guaranteethat the following statement is sufficient to prove that (4.9) holds for everychoice of function ψ . For every integer r ≥ µ ∞ [ | q | = r +1 R ψ ( q ) ≥ a ∗ c ( x ) / . Note that this statement is in terms of the constructed function ψ . To prove(4.12) we will show that there cannot exist a natural number t such thatit fails to hold when r = k t . Assume that such a t exists and consider thecollection of rectangles defined by R t := R ( ψ , t ) := k t [ | q | = k to +1 R ψ ( q ) for t = t + 1 , t + 2 , . . . . We will demonstrate that the measure of the set R t is unbounded as t increases and in doing so reach a contradiction, as each R t is containedin [0 , . We will do this by estimating the size of a suitable sum of themeasure of set differences of the form R t +1 \ R t .By construction each R t +1 is obtained from R t by adding 2( k t +1 − k t )new rectangles to those of R t . These new rectangles are centred at thepoints q x for which k t < | q | ≤ k t +1 . To estimate µ ( R t +1 \ R t ) we will findan upper bound to the number of the new rectangles that intersect anyexisting rectangle of R t . In particular, we will find an upper bound to thecardinality of the set J t +1 ∩ t , where J t +1 denotes the set of points q x for which k t < | q | ≤ k t +1 and t := k t [ | q | = k to +1 R ψ ( q ) for t = t + 1 , t + 2 , . . . . This will suffice as ψ is non-increasing. Before proceeding we first noticethat, since the vector x was chosen from Bad ( i, j ), if q x and q ′ x are membersof J t +1 then(4.13) max n k qx − q ′ x k /i , k qx − q ′ x k /j o ≥ c ( x ) | q − q ′ | ≥ c ( x )2 k t +1 , providing that the integers q and q ′ are distinct.The collection t can be partitioned into two exhaustive subcollections(which we will assume without loss of generality are non-empty). Recallingthat a ∗ := 2 − / min { i,j } , define (1) t := [ R ψ ( q ) , where the union runs over all non-zero q with k t < | q | ≤ k t such that2 ψ ( | q | ) < a ∗ c ( x )2 k t +1 . In turn, let (2) t := [ R ψ ( q ) , where this time the union runs over q with k t < | q | ≤ k t such that2 ψ ( | q | ) ≥ a ∗ c ( x )2 k t +1 . The intersections J t +1 ∩ (1) t and J t +1 ∩ (2) t will now be dealt withindependently.The subcollection (1) t consists of rectangles of sidelengths2(2 ψ ( | q | )) i and 2(2 ψ ( | q | )) j .I.D.A. & BADLY APPROXIMABLE SETS 17 respectively and both2 (2 ψ ( | q | )) i < (cid:18) c ( x )2 k t +1 (cid:19) i and 2 (2 ψ ( | q | )) j < (cid:18) c ( x )2 k t +1 (cid:19) j . This follows upon noticing that max { a i ∗ , a j ∗ } = 1 /
2. Thus, statement (4.13)implies at most one element of J t +1 can lie in each rectangle of (1) t andso J t +1 ∩ (1) t contains at most 2( k t − k t ) < k t elements.Estimating the cardinality of J t +1 ∩ (2) t requires more work and weargue as follows. If a point γ lies in the subcollection (2) t then it mustlie in a rectangle of the form R ψ ( q ) ⊆ (2) t for some integer q with k t < | q | ≤ k t . This rectangle must have respective sidelengths 2(2 ψ ( | q | )) i and 2(2 ψ ( | q | )) j and by definition we have2 (2 ψ ( | q | )) i ≥ (cid:18) a ∗ c ( x )2 k t +1 (cid:19) i and 2 (2 ψ ( | q | )) j ≥ (cid:18) a ∗ c ( x )2 k t +1 (cid:19) j . It is now clear that there must exist a point y ( γ ) ∈ R ψ ( q ) such that γ iscontained in a subrectangle, say S ( γ ), of R ψ ( q ) centred at y ( γ ) and ofsidelengths ( a ∗ c ( x ) / k t +1 ) i and ( a ∗ c ( x ) / k t +1 ) j respectively. The fact thatmax { a i ∗ , a j ∗ } = 1 /
2, twinned with equation (4.13), once more guaranteesthat only one point of J t +1 may lie in any subrectangle of this type. More-over, any two such subrectangles containing respective points q x and q ′ x ,both in J t +1 , must be disjoint. Thus, the cardinality of J t +1 ∩ (2) t cannotexceed µ ( (2) t ) /µ ( S ( γ )). We estimate the size of µ ( (2) t ) by utilising thefollowing lemma. Lemma 4.3.
For every t = t + 1 , t + 2 , . . . , µ ( t ) ≤ µ ( R t ) . Proof of Lemma 4.3.
For s ∈ N , let R s := k to + s [ | q | = k to +1 R ψ ( q ) and s := k to + s [ | q | = k to +1 R ψ ( q ) . To prove Lemma 4.3 it suffices to show that µ ( s ) ≤ µ ( R s ) for all s .We proceed by induction. If s = 1, then µ ( R ) = 2 ψ i ( k t + 1) · ψ j ( k t + 1) = 4 ψ ( k t + 1) . Further, µ ( ) = 2(2 ψ ( k t + 1)) i · ψ ( k t + 1)) j = 2 · ψ ( k t + 1) = 2 µ ( R )and the statement holds. Next, assume the hypothesis holds when s = s ′ and define a transforma-tion T on the torus [0 , by T ( γ ) := (cid:0) i γ , j γ (cid:1) ∀ γ ∈ [0 , . For any subset D ⊆ [0 , , we denote by T ( D ) the set of all points T ( γ )where γ ∈ D . Let D s ′ +1 := R s ′ +1 \ R s ′ , then, since by definition ψ doesnot exceed a ∗ ( i, j ) / µ ( T ( D s ′ +1 )) = 2 i · j · µ ( D s ′ +1 ) = 2 µ ( D s ′ +1 ) . It is also clear that s ′ +1 = s ′ ∪ T ( D s ′ +1 ) , from which it follows that µ ( s ′ +1 ) = µ ( s ′ ∪ T ( D s ′ +1 )) ≤ µ ( s ′ ) + µ ( T ( D s ′ +1 )) ≤ µ ( R s ′ ) + 2 µ ( D s ′ +1 ) (by assumption and (4.14) resp.)= 2 µ ( R s ′ ∪ D s ′ +1 ) (since R s ′ and D s ′ +1 are disjoint)= 2 µ ( R s ′ +1 ) , as required. (cid:3) We return to our calculation. Assuming as we are that statement (4.12)is false, Lemma 4.3 now yields that µ ( (2) t ) ≤ µ ( t ) ≤ µ ( R t ) < a ∗ c ( x ) / . Thus, J t +1 ∩ (2) t ) ≤ µ ( (2) t ) µ ( S ( γ )) < a ∗ c ( x )4 ( a ∗ c ( x ) / k t +1 ) i + j = k t +1 µ ( R t +1 \ R t ), we can now write downan upper bound for the number of rectangles added to R t to make R t +1 thatdo intersect existing rectangles of R t . Indeed, this number cannot exceed(4.15) J t +1 ∩ t ) ≤ k t + k t +1 / , which follows upon noticing that J t +1 ∩ t = ( J t +1 ∩ (1) t ) ∪ ( J t +1 ∩ (2) t ) . To complete our arguement we require one final piece of notation. Let L t +1 := { q ∈ Z =0 : q x ∈ J t +1 , q x / ∈ t } . .I.D.A. & BADLY APPROXIMABLE SETS 19 The integers q ∈ L t +1 each correspond to a rectangle of R t +1 that does notintersect any rectangle of R t . So, by (4.15) L t +1 ) ≥ k t +1 − k t ) − (2 k t + k t +1 / − /k − / k t +1( ) > (2 − − / k t +1 = k t +1 / . (4.16)We will now estimate µ ( R t +1 \ R t ) by considering the inclusion(4.17) R t +1 \ R t ⊃ [ q ∈ L t +1 R ψ ( q ) . The rectangles R ψ ( q ) in the above union have sidelengths 2 ψ i ( | q | ) and2 ψ j ( | q | ) respectively. Further, if q, q ′ ∈ L t +1 then k t < | q | , | q ′ | ≤ k t +1 andso(4.18) max n k qx − q ′ x k /i , k qx − q ′ x k /j o ( ) ≥ c ( x )2 k t +1 . Recall that ψ is constant on each L t +1 by definition, taking the value ψ ( k t +1 ), and also that ψ ( r ) ≤ a ∗ c ( x )2 | r | . Therefore, we have both2 ψ i ( | q | ) = 2 ψ i ( k t +1 ) < (cid:18) c ( x )2 k t +1 (cid:19) i and 2 ψ j ( | q | ) = 2 ψ j ( k t +1 ) < (cid:18) c ( x )2 k t +1 (cid:19) j . Combining these inequalities with statement (4.18) yields that the rectan-gles R ψ ( q ) on the RHS of (4.17) are disjoint. Hence, µ ( R t +1 \ R t ) ≥ X q ∈ L t +1 µ ( R ψ ( q ))= 2 X q ∈ L t +1 ψ ( | q | ) ( ) > k t +1 ψ ( k t +1 ) > k t +1 − k t ) ψ ( k t +1 )= k t +1 X | q | = k t +1 ψ ( k t +1 )= k t +1 X | q | = k t +1 ψ ( | q | ) . Finally, ψ is divergent; i.e. ∞ X | q | =1 ψ ( | q | ) = ∞ , whence P t>t µ ( R t +1 \ R t ) = ∞ . Since R t ⊆ R t +1 for any t > t , thisimplies that µ ( R t ) → ∞ as t → ∞ . However, each set R t is contained in[0 , and so a contradiction is reached. This means the assumption that(4.12) fails for some r = k t o is indeed false, and consequently µ (cid:0) W x ( i,j ) ( ψ ) (cid:1) > ψ ∈ D as desired.To complete the proof of Proposition 4.2 we must now show if x ∈ Bad ( i, j ) then µ (cid:0) W x ( i,j ) ( ψ ) (cid:1) = 1for every ψ ∈ D . Our method will be through the application of two lemmas,the first of which is due to Kurzweil ([21, Lemma 13]). Lemma 4.4 (Kurzweil) . Let U and V be subsets of [0 , . If µ ( U ) > and V is dense in [0 , then µ ( U ⊕ V ) = 1 , where U ⊕ V := { u + v ( mod
1) : u ∈ U, v ∈ V } . Lemma 4.5.
For every ψ ∈ D and for every natural number s we have ∞ X r =1 ψ ( sr ) = ∞ . .I.D.A. & BADLY APPROXIMABLE SETS 21 Proof of Lemma 4.5.
Suppose s ≥ ψ (0) := ψ (1). Consider the s -subseries P ∞ r =0 ψ ( sr + k ) for each k = 0 , . . . , s −
1. Everyterm ψ ( r ′ ), r ′ ∈ N , appears exactly once in exactly one s -subseries. If every s -subseries had a finite sum then the original series P ∞ r =1 ψ ( r ) would alsohave a finite sum (precisely equal to the sum of the sums of the s -subseries).Since the original series does not have a finite sum, at least one of the s -subseries must diverge, say P ∞ r =0 ψ ( sr + k ) = ∞ . Since ψ is decreasing ψ ( sr ) ≥ ψ ( sr + k ) and so P ∞ r =0 ψ ( sr ) = ∞ and Lemma 4.5 holds. (cid:3) Returning to the proof of Proposition 4.2, fix a divergent approximatingfunction ψ and a vector x ∈ Bad ( i, j ). Once again, we will refine ψ beforeproceeding. Firstly, we will construct a function ψ ∈ D such that(4.19) lim r →∞ (cid:18) ψ ( r ) ψ ( r ) (cid:19) = 0 . Let r = 0 and choose r ≥ P r r =1 ψ ( r ) ≥ { r k } ∞ k =0 such that for each k (4.20) r k X r = r k − +1 ψ ( r ) ≥ k. This is always possible since P ∞ r =1 ψ ( r ) diverges, so the partial sums fromany starting point must tend to infinity. Next, define c r := 1 / √ k if r k −
The proof of Theorem 2.2 makes use of the framework developed in [19].This framework was specifically designed to provide dimension results fora broad range of badly approximable sets. In this section we show that
Bad x ( i, j ) falls into this category when x is chosen from Bad ( i, j ). First,we provide a simplification of the framework tailored to our needs.Let R := { R α ⊂ R : α ∈ J } be a family of subsets R α of R indexedby an infinite countable set J . We will refer to the sets R α as resonantsets . Furthermore, it will be assumed that each resonant set takes the formof a cartesian product; i.e., that each set R α can be split into the images R α,t ⊂ R , t = 1 ,
2, of its two projection maps along the two coordinateaxis. Next, let β : J → R > : α β α be a positive function on J suchthat the number of α ∈ J with β α bounded above is finite. Thus, as α runs through J the function β α tends to infinity. Also, for t = 1 ,
2, let ρ t : R > → R > : r ρ t ( r ) be any real, positive, decreasing functionsuch that ρ t ( r ) → r → ∞ . We assume that either ρ ( r ) ≥ ρ ( r ) or ρ ( r ) ≥ ρ ( r ) for large enough r . Finally, for each resonant set R α define arectangular neighbourhood F α ( ρ , ρ ) by F α ( ρ , ρ ) := (cid:8) x ∈ R : | x t − R α,t | ≤ ρ t ( β α ) for t = 1 , (cid:9) , where | x t − R α,t | := inf a ∈R α,t | x t − a | .We now introduce the general badly approximable set to which the re-sults of [19] relate. Define Bad ( R , β, ρ , ρ ) to be the set of x ∈ [0 , forwhich there exists a constant c ( x ) > x / ∈ c ( x ) F α ( ρ , ρ ) ∀ α ∈ J. That is, x ∈ Bad ( R , β, ρ , ρ ) if there exists a constant c ( x ) > α ∈ J | x t − R α,t | ≥ c ( x ) ρ t ( β α ) ( t = 1 , . The aim of the framework is to determine conditions under which theset
Bad ( R , β, ρ , ρ ) has full Hausdorff dimension. With this in mind, webegin with some useful notation. For any fixed integers k > n ≥ F n := (cid:8) x ∈ [0 , : | x t − c t | ≤ ρ t ( k n ) for each t = 1 , (cid:9) to be the generic closed rectangle in [0 , with centre c := ( c , c ) andof side lengths given by 2 ρ ( k n ) and 2 ρ ( k n ) respectively. Next, for any θ ∈ R > , let θF n := (cid:8) x ∈ [0 , : | x t − c t | ≤ θρ t ( k n ) for each t = 1 , (cid:9) denote the rectangle F n scaled by θ . Finally, let J ( n ) := (cid:8) α ∈ J : k n − ≤ β α < k n (cid:9) . The following statement is a simplification of Theorem 2 of [19], madepossible by the properties of the 2-dimensional Lebesgue measure µ . Theorem KTV (2006).
Let k be sufficiently large. Suppose there existssome θ ∈ R > such that for any n ≥ and any rectangle F n there existsa collection C ( θF n ) of disjoint rectangles θF n +1 contained within θF n suchthat (5.1) C ( θF n ) ≥ κ µ ( θF n ) µ ( θF n +1 ) and { θF n +1 ⊂ C ( θF n ) : R α ∩ θF n +1 = ∅ for some α ∈ J ( n + 1) } (5.2) ≤ κ µ ( θF n ) µ ( θF n +1 ) , where < κ < κ are absolute constants independent of k and n . Further-more, suppose (5.3) dim ( ∪ α ∈ J R α ) < , then dim ( Bad ( R , β, ρ , ρ )) = 2 . We can now prove Theorem 2.2.
Proof of Theorem 2.2.
Fix two positive reals i, j with i + j = 1 and some x ∈ Bad ( i, j ). It is once more assumed that i, j >
0, for in this case thetheorem would otherwise follow immediately from Corollary 1 of [4]. Withreference to the above framework, set J := { q ∈ Z =0 } , α := q ∈ J, R α := R q = (cid:8) q x + p : p ∈ Z (cid:9) β α := β q = | q | , ρ ( r ) := 1 /r i and ρ ( r ) := 1 /r j . By design we then have
Bad ( R , β, ρ , ρ ) = Bad x ( i, j )and so the proof is reduced to showing that the conditions of Theorem KTVare satisfied.For k > m ≥
1, let F m be a generic closed rectangle with centre in[0 , and of side lengths 2 k − mi and 2 k − mj respectively . For k sufficientlylarge and any θ ∈ R > it is clear that there exists a collection C ( θB m )of closed rectangles 2 θF m +1 within θF m each of side lengths 4 θk − ( m +1) i and .I.D.A. & BADLY APPROXIMABLE SETS 25 θk − ( m +1) j respectively. Moreover, the number of rectangles in this collectionexceeds (cid:22) θk − mi θk − ( m +1) i (cid:23) × (cid:22) θk − mj θk − ( m +1) j (cid:23) . Here, the notation ⌊ . ⌋ denotes the integer part. For large enough k theabove is strictly positive and is bounded below by12 (cid:18) θk − mi θk − ( m +1) i (cid:19) × (cid:18) θk − mj θk − ( m +1) j (cid:19) = 116 (cid:18) θ k − m ( i + j ) θ k − ( m +1)( i + j ) (cid:19) = 116 µ ( θF m ) µ ( θF m +1 ) . Hence, inequality (5.1) holds with κ := 1 / C ( θF m ) is satisfied. To this end, we fix m ≥ J ( m + 1); i.e., choosetwo integers q and q ′ such that(5.4) k m ≤ | q ′ | < | q | < k m +1 . Associated with the integers q and q ′ are the resonant sets R q and R q ′ ,whose elements take the form q x + p and q ′ x + p ′ respectively (for some p , p ′ ∈ Z ). Consider the minimum distance between a point in R q and onein R q ′ . For t = 1 , | ( qx t + p t ) − ( q ′ x t + p ′ t ) | = | ( q − q ′ ) x t + p t − p ′ t |≥ k ( q − q ′ ) x t k . Since x ∈ Bad ( i, j ) either k ( q − q ′ ) x k ≥ (cid:18) c ( x ) | q − q ′ | (cid:19) i ( ) > (cid:18) c ( x )2 k m +1 (cid:19) i or k ( q − q ′ ) x k ≥ (cid:18) c ( x ) | q − q ′ | (cid:19) j ( ) > (cid:18) c ( x )2 k m +1 (cid:19) j . Therefore, if we set θ := 12 min ((cid:18) c ( x )2 k (cid:19) i , (cid:18) c ( x )2 k (cid:19) j ) then the rectangle θF m has respective side lengths2 θk − mi = min ((cid:18) c ( x )2 k (cid:19) i , (cid:18) c ( x )2 k (cid:19) j ) k − mi ≤ (cid:18) c ( x )2 k m +1 (cid:19) i and 2 θk − mj = min ((cid:18) c ( x )2 k (cid:19) i , (cid:18) c ( x )2 k (cid:19) j ) k − mj ≤ (cid:18) c ( x )2 k m +1 (cid:19) j . So, for any two integers q, q ′ of distinct moduli in J ( m + 1), if a memberof R q lies in θF m then no members of R q ′ may lie in θF m . Only one pointof R q may lie in θF m (since µ ( θF m ) <
1) and so only two points over allpossible resident sets may lie in any rectangle θF m ; those corresponding to q and − q . Hence, { θF m +1 ⊂ C ( θF m ) : R q ∩ θF m +1 = ∅ for some q ∈ J ( m + 1) } ≤ , which for large enough k is certainly less than k
32 = 132 µ ( θF m ) µ ( θF m +1 ) . So, with θ as defined above and with κ := 1 / < κ , the collection C ( θF m )satisfies inequality (5.2).Finally, note that the family R of resonant sets takes the form of acountable number of countable sets and sodim ( ∪ q ∈ J R q ) = 0and inequality (5.3) trivially holds. Thus, the conditions of Theorem KTVare satisfied and the theorem follows. (cid:3) Appendix
We conclude the paper by proving a general result implying Theorems1.5 & 3.3 as stated in the main body of the paper. The result is an exten-sion of Cassels’ inhomogeneous Khintchine-type theorem [6, Chapter VII,Theorem II]. The proof is a modification of Cassels’ original argument andalso borrows ideas from the work of Gallagher.
Theorem 6.1.
For any sequence { A q } q ∈ N of measurable subsets of [0 , d let A denote the set of all pairs ( x , γ ) ∈ [0 , d × [0 , d for which there existsinfinitely many q ∈ N and p ∈ Z d such that (6.1) q x − γ − p ∈ A q . Then, µ d ( A ) := , ∞ X r =1 µ d ( A r ) < ∞ , , ∞ X r =1 µ d ( A r ) = ∞ , where µ s denotes s -dimensional Lebesgue measure.Proof. We begin by considering the case in which the sum P ∞ r =1 µ d ( A r )converges. Fix γ ∈ [0 , d . For each natural number q a vector x satisfying(6.1) uniquely determines the integral vector p in such a way that | p | < q . .I.D.A. & BADLY APPROXIMABLE SETS 27 Therefore, the measure of the set of all x ∈ [0 , d that satisfy (6.1) for each q is given by µ d [ p ∈ [0 , q ) d ( A q ⊕ γ ) ⊕ p q = X p ∈ [0 , q ) d µ d (cid:18) ( A q ⊕ γ ) ⊕ p q (cid:19) , since the union is disjoint. The dilation property of µ d yields that this isequivalent to q − d X p ∈ [0 , q ) d µ d (( A q ⊕ γ ) ⊕ p ) = q − d q d · µ d ( A q ⊕ γ ) = µ d ( A q ) , by the translational invariance of µ d . Now, if P ∞ r =1 µ d ( A r ) < ∞ , then forany ǫ > q ≥ Q has measure atmost P q ≥ Q µ d ( A q ) < ǫ for large enough Q . In particular, the set of x withinfinitely many solutions to (6.1) has measure at most ǫ . This completes theproof of the convergence case.Let us now assume that the sum P ∞ r =1 µ d ( A r ) diverges. Define the func-tion α q : R d → R for each natural number q as follows. Let α q ( x ) : = ( , ∃ p ∈ Z d s.t. x − p ∈ A q . , otherwise . It is clear that each α q is measurable since it is equivalent to the character-istic function of a countable union of measurable sets in R d . Next, for everynatural number Q define the function A Q : [0 , d × [0 , d → R by A Q ( x , γ ) : = X q ≤ Q α q ( q x − γ ) . We wish to verify that A Q is measurable. To that end, we introduce thefollowing lemma, which is a generalisation of a well known result in measuretheory and follows via simple modification of the classical proof (see forexample [28, Chapter 2, Proposition 3.9]). Lemma 6.2. If f is a measurable function on R d then it follows that thefunction F q ( x , γ ) := f ( q x − γ ) is measurable on R d × R d for every naturalnumber q . Since α q is finite valued (and finite sums of finite valued measurablefunctions are measurable functions) Lemma 6.2 implies that A Q is indeedmeasurable on [0 , d × [0 , d . Furthermore, by construction, it is apparentthat A Q ( x , γ ) is simply the number of natural q with q ≤ Q such that q x − γ − p ∈ A q for some p ∈ Z d . Hence, to complete the proof of Theorem 6.1 it suffices to show A Q ( x , γ ) →∞ almost everywhere as Q → ∞ . We will hereafter consider A Q as a randomvariable in a probability space with probability measure µ d .For any positive measurable function f : [0 , d × [0 , d → R d ≥ we denotethe expectation of f by E ( f ) := Z [0 , d Z [0 , d f ( x , γ ) d x d γ . If the variance V ( f ) := E ( f ) − E ( f ) of f is finite then the famous Paley-Zygmund inequality (see for example [16, Ineq. II, p.8]) states that µ d ( { ( x , γ ) : f ( x , γ ) ≥ ǫE ( f ) } ) ≥ (1 − ǫ ) ( E ( f )) E ( f ) , for any sufficiently small ǫ >
0. We will use this inequality to reach ourdesired conclusion.Before applying the Paley-Zygmund inequality to A Q we must show that V ( A Q ) is finite. It suffices to show that both E ( A Q ) and E (( A Q ) ) are finite.To do this we require the following lemma [6, Chapter VII, Lemma 3]. Lemma 6.3 (Cassels) . Let α be a measurable function of period one of thevariable x ∈ R d . Then, Z [0 , d α ( q x + γ ) d x = Z [0 , d α ( x ) d x , for any vector γ ∈ R d and any integer q = 0 . We note that α q is of period one and so E ( A Q ) = Z [0 , d Z [0 , d A Q ( x , γ ) d x d γ = X q ≤ Q Z [0 , d Z [0 , d α q ( q x − γ ) d x d γ Lem. 6.3 = X q ≤ Q Z [0 , d Z [0 , d α q ( x ) d x d γ = X q ≤ Q Z [0 , d Z [0 , d χ A q ( x ) d x d γ = X q ≤ Q µ d ( A q ) , (6.2) .I.D.A. & BADLY APPROXIMABLE SETS 29 which is indeed finite. Further, E (( A Q ) ) = Z [0 , d Z [0 , d ( A Q ( x , γ )) d x d γ = X q,r ≤ Q Z [0 , d Z [0 , d α q ( q x − γ ) α r ( r x − γ ) d x d γ = X q,r ≤ Q Z [0 , d Z [0 , d α r − s ( − γ ′ ) α r ( s x ′ − γ ′ ) d x ′ d γ ′ , via the change of variables x ′ := x , γ ′ := γ − q x and s := r − q . Here,the range of x ′ and γ ′ can both be taken as [0 , d since the function α q isperiodic. Let A ( r,s ) ( x ′ , γ ′ ) : = Z [0 , d Z [0 , d α r − s ( − γ ′ ) α r ( s x ′ − γ ′ ) d x ′ d γ ′ . Then, if r = q then s = 0 and we have A ( r,s ) ( x ′ , γ ′ ) = Z [0 , d Z [0 , d ( α q ( − γ ′ )) d x ′ d γ ′ = Z [0 , d Z [0 , d α q ( − γ ′ ) d x ′ d γ ′ = µ d ( A q ) . However, if r = q then s = 0 and we get A ( r,s ) ( x ′ , γ ′ ) = Z [0 , d α r − s ( − γ ′ ) d x ′ Z [0 , d Z [0 , d α r ( s x ′ − γ ′ ) d x ′ d γ ′ Lem. 6.3 = µ d ( A r − s ) Z [0 , d Z [0 , d α r ( x ′ ) d x ′ d γ ′ = µ d ( A q ) µ d ( A r ) . These equivalences yield that E (( A Q ) ) = X q,r ≤ Q A ( r,s ) ( x ′ , γ ′ )= X q ≤ Q µ d ( A q ) + X q,r ≤ Q : q = r µ d ( A q ) µ d ( A r ) ≤ X q ≤ Q µ d ( A q ) + X q ≤ Q µ d ( A q ) ! ≤ (1 − ǫ ) − X q ≤ Q µ d ( A q ) ! = (1 − ǫ ) − ( E ( A Q )) , for any sufficiently small ǫ > Q (because P q ≤ Q µ d ( A q ) →∞ as Q → ∞ by assumption). Note that the final bound is finite as required.In view of the Paley-Zygmund inequality we have that µ d ( ( x , γ ) : A Q ( x , γ ) ≥ ǫ X q ≤ Q µ d ( A q ) )! ≥ (1 − ǫ ) ≥ − ǫ. Finally, since A Q increases monotonically with Q , we have that A Q ( x , γ ) →∞ in [0 , d × [0 , d except on a set of measure at most 4 ǫ . This completesthe proof as the choice of ǫ is arbitrary. (cid:3) Acknowledgements.
The author would like to thank Sanju Velani for hisenthusiastic encouragement and support, and for introducing him to theproblems at hand. He would also like to thank the referees for their usefulremarks and Simon Eveson for his helpful suggestions. This research wasfunded by the ESPRC.
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