Optimality of two inequalities for exponents of Diophantine approximation
aa r X i v : . [ m a t h . N T ] F e b OPTIMALITY OF TWO INEQUALITIES FOR EXPONENTS OFDIOPHANTINE APPROXIMATION
JOHANNES SCHLEISCHITZ
Dedicated to the 50th birthday of Yann Bugeaud
Abstract.
We investigate two inequalities of Bugeaud and Laurent, each involvingtriples of classical exponents of Diophantine approximation associated to ξ ∈ R n . Weprovide a complete description of parameter triples that admit equality for suitable ξ ,which turns out rather surprising. For n = 2 our results agree with work of Laurent.Moreover, we establish lower bounds for the Hausdorff and packing dimensions of theinvolved ξ , and in special cases we can show they are sharp. Proofs are based on thevariational principle in parametric geometry of numbers, we enclose sketches of associ-ated combined graphs (templates) where equality is feasible. A twist of our constructionprovides refined information on the joint spectrum of the respective exponent triples. Keywords : exponents of Diophantine approximation, parametric geometry of numbersMath Subject Classification 2010: 11J13, 11J82 Introduction
Classical exponents of approximation.
Let n ≥ ξ =( ξ , . . . , ξ n ) ∈ R n with { , ξ , . . . , ξ n } linearly independent over Q . Let the (possiblyinfinite) exponents of approximation ω and b ω resp. be defined as the suprema of reals u so that the system(1) 1 ≤ x ≤ X, max ≤ i ≤ n | xξ i − y i | ≤ X − u has a solution in integer vectors ( x, y , . . . , y n ) for arbitrarily large and all large X , re-spectively. Let ω ∗ , b ω ∗ be the supremum of v so that1 ≤ max ≤ i ≤ n | a i | ≤ X, | a + a ξ + · · · + a n ξ n | ≤ X − v has a solution in integers a i for arbitrarily large X and all large X , respectively. Byvariants of Dirichlet’s Theorem, for any ξ ∈ R n we have(2) ∞ ≥ ω ≥ b ω ≥ n , ∞ ≥ ω ∗ ≥ b ω ∗ ≥ n. Middle East Technical University, Northern Cyprus Campus, Kalkanli, G¨[email protected] ; [email protected].
Two inequalities by Bugeaud and Laurent.
Let n ≥ ξ as above satisfies the estimates(BL1) ω ≥ ( b ω ∗ − ω ∗ (( n − b ω ∗ + 1) ω ∗ + ( n − b ω ∗ and(BL2) ω ∗ ≥ ( n − ω + b ω + n − − b ω . In view of (2) they imply Khintchine’s transference inequalities [9](3) ω ∗ ≥ nω + n − , ω ≥ ω ∗ ( n − ω ∗ + n . These are known to be sharp for any parameter pairs induced by ω ∈ [1 /n, ∞ ] resp. ω ∗ ∈ [ n, ∞ ]. Moreover, combining (BL1), (BL2) implies non-trivial relations between ω and b ω , and likewise between ω ∗ and b ω ∗ . For n = 2, they become(4) ω ≥ b ω − b ω , ω ∗ ≥ b ω ∗ − b ω ∗ , already known by Jarn´ık [8] and are again sharp for any parameter pairs induced by ω ∈ [1 /n, ∞ ] resp. ω ∗ ∈ [ n, ∞ ]. For n >
2, the implied relations turn out to be no longerbest possible. Marnat and Moshchevitin [12] settled the sharp estimates conjecturedby Schmidt and Summerer [23] along their discussion of the ”regular graph”, who gaveproofs for n = 3 themselves. See also Schmidt and Summerer [22], Moshchevitin [13] andRivard-Cooke’s thesis [14].2. Main result: Description of equality cases
For n = 2 all estimates (BL1), (BL2) are sharp by a result of Laurent [10], as alreadypointed out in [2]. For general n ≥
2, when ω ∗ ≥ n, b ω ∗ = n resp. ω ≥ /n, b ω = 1 /n inequalities (BL1) resp. (BL2) simplify to (3) and are sharp, as noticed in Section 1.2.Other than that, when n > ω ∗ , b ω ∗ , ω ) ⊆ ( R ∪ {∞} ) induced by ξ ∈ R n with thehypersurface in ( R ∪ {∞} ) induced by equality in (BL1). For given n ≥ x ≥ n , define ρ ( n, x ) = x ( n − x + n , and ρ ( n, x ) = (2 n − x + (2 n − − p (4 n − x + 1) x − p (4 n − x + 1 + 12(( n − x + (2 n − n + 3) x + n − n ) . By taking limits we put ρ ( n, ∞ ) = 1 / ( n −
1) and ρ ( n, ∞ ) = 1 / ( n − / ∞ . Observe ρ ( n, n ) = ρ ( n, n ) = 1 /n . Our results will involve Hausdorff andpacking dimension, see [5] for an introduction. PTIMALITY OF TWO INEQUALITIES 3
Theorem 2.1.
Let n ≥ be an integer. Then precisely for triples ( w ∗ , b w ∗ , w ) ⊆ ( R ∪{∞} ) that can be parametrized as (5) w ∗ ∈ [ n, ∞ ] , w ∈ [ ρ ( n, w ∗ ) , ρ ( n, w ∗ )] , b w ∗ = w ∗ ( w + 1) w ∗ − ( n − ww ∗ − ( n − w there is ξ ∈ R n which induces equality in (BL1) and (6) ω ∗ = w ∗ , b ω ∗ = b w ∗ , ω = w. In fact, for each admissible parameter triple ( w ∗ , b w ∗ , w ) as in (5) , the set of ξ inducing (6) has Hausdorff dimension at least n − and packing dimension at least n − /n . The identity in (5) is a reformulation of equality in (BL1) and its explicit statementis purely conventional. The lower bound ρ reflects identity in the right estimate of (3),so it is as small as it can possibly be. On the other hand, the surprising bound ρ isstrictly smaller than the bound ( w ∗ − n + 1) /n from the left estimate in (3), unless intrivial cases. Thus, by the optimality of Khintchine’s estimates, when w ∗ ∈ ( n, ∞ ] and w ∈ ( ρ , ( w ∗ − n + 1) /n ], there are ξ whose associated exponents satisfy ω ∗ = w ∗ , ω = w but there is no ξ with additional equality in (BL1). We remark that if w = ρ , then as w ∗ → ∞ also b w ∗ → ∞ , and w → / ( n − ω = ρ ( n, ω ∗ ) and equalityin (SS1) holds, we can evaluate b ω as well. We omit stating the complicated expression,but remark that b ω → / ( n −
1) as ω ∗ → ∞ . As w ∗ → n from above, both ρ , ρ tend to1 /n as it needs to be. For refinements of the metrical claim see Section 4.2 below.For n = 2 the claim agrees with the findings of Laurent [10] discussed above. For n = 2and ω = ρ we have equality in the inequalities of (4), thereby we obtain a regular graphas mentioned in Section 1.2. For n >
2, no regular graph can appear upon identity in(BL1), unless in the trivial case w ∗ = b w ∗ = n, w = 1 /n .Our results suggest that solutions to the following questions are in reach. Problem 1.
For given ω ∗ and ω > ρ ( n, ω ∗ ) find sharp estimates improving (BL1).Ideally, determine the spectrum of ( ω ∗ , b ω ∗ , ω ) ⊆ ( R ∪ {∞} ) .We turn towards the dual estimates (BL2). For n ≥ x ≥ /n , define τ ( n, x ) = nx + n − , and τ ( n, x ) = x n −
12 + p x ( x + 4 n − ! x + p x ( x + 4 n − n − . By taking limits we extend it to τ j ( n, ∞ ) = ∞ for j = 1 ,
2. Our result reads as follows.
Theorem 2.2.
Let n ≥ be an integer. Then precisely for triples ( w, b w, w ∗ ) ⊆ ( R ∪{∞} ) that can be parametrized by the properties (7) w ∈ [1 /n, ∞ ] , w ∗ ∈ [ τ ( n, w ) , τ ( n, w )] , b w = w ∗ − ( n − w − n + 21 + w ∗ JOHANNES SCHLEISCHITZ there is ξ ∈ R n which induces equality in (BL2) and (8) ω = w, b ω = b w, ω ∗ = w ∗ . For each suitable parameter triple ( w, b w, w ∗ ) , the set of associated ξ inducing (8) haspacking dimension at least / , and positive Hausdorff dimension if w < ∞ . Analogous remarks as for Theorem 2.1 apply. The lower bound τ reflects equality inthe left estimate of (3), whereas τ is strictly smaller than the value induced by equalityin the right inequality of (3), unless if ω = ω ∗ = n . Let w ∗ = τ . Then, as w → ∞ wehave b w → w ∗ → ∞ , where the latter agrees with the obvious estimate ω ∗ ≥ ω .Note that 1 is the largest value b ω can attain. Moreover if ω ∗ = τ ( n, ω ) we may evaluate b ω ∗ . If ω ∗ = τ ( n, ω ) then again as ω → ∞ also b ω ∗ → ∞ , and iff n = 2 this leads to theregular graph. We formulate the analogous questions to Problem 1. Problem 2.
For ω ∗ > τ ( n, ω ), find sharp estimates improving (BL2). Ideally, determinethe spectrum of ( ω, b ω, ω ∗ ) ⊆ ( R ∪ {∞} ) .As partial results to Problems 1, 2, for any triples satisfying(9) w ∗ ∈ [ n, ∞ ] , w ∈ [ ρ ( n, w ∗ ) , ρ ( n, w ∗ )] , b w ∗ ∈ (cid:20) n, w ∗ ( w + 1) w ∗ − ( n − ww ∗ − ( n − w (cid:21) resp.(10) w ∈ [1 /n, ∞ ] , w ∗ ∈ [ τ ( n, w ) , τ ( n, w )] , b w ∈ (cid:20) n , w ∗ − ( n − w − n + 21 + w ∗ (cid:21) for suitable ξ ∈ R n we still get (6) resp. (8). Clearly the ranges for b ω ∗ , b ω are optimal.See Section 5.7 for the proof. Problems 1, 2 can be considered partial problems towardsfinding the entire spectrum in R n +2 of all extremal values of successive minima exponents,which is wide open for n >
2. See Section 3.1 for details.We outline the rest of the paper. In Section 3 we introduce parametric geometry ofnumbers in the notion of Schmidt and Summerer [22] and formulate a special case thevariational principle by Das, Fishman, Simmons, Urba´nski [3], [4] within this framework.We append relations to notions of Roy [16], [17] and Schmidt, Summerer [25]. In Section 4we reformulate Theorems 2.1, 2.2 into this language, and refine the metrical claims. InSections 5, 6 these claims and (9), (10) are proved using the prerequisites from [3], [4].It is worth noting that everything except from the metrical claims can be alternativelyobtained from Roy [16], [17] in place of [3], [4], see Section 3.2 for details. Finally weclose with some remarks interconnecting our work with [2], [17] in Section 7.3.
Parametric geometry of numbers
Parametric functions and their extremal values.
Let us interpret the simulta-neous approximation problem (1) as a parametric successive minima problem. For q >
0a parameter, let K ( q ) ⊆ R n +1 be the box of points ( z , z , . . . , z n ) that satisfy | z | ≤ e nq , max ≤ i ≤ n | z i | ≤ e − q . PTIMALITY OF TWO INEQUALITIES 5
Further let Λ = Λ ξ be the lattice consisting of all points of the form { ( x, ξ x − y , . . . , ξ n x − y n ) : x, y i ∈ Z } . Denote by λ ( q ) , . . . , λ n +1 ( q ) the successive minima of K ( q ) with respectto Λ, to obtain functions of q . For 1 ≤ j ≤ n + 1, derive ϕ j ( q ) = log λ j ( q ) /q and definethe lower and upper limits ϕ j = lim inf q →∞ ϕ j ( q ) , ϕ j = lim sup q →∞ ϕ j ( q ) . These quantities lie within the interval [ − n, ω )( n + ϕ ) = (1 + b ω )( n + ϕ ) = n + 1 , and(T2) (1 + ω ∗ )(1 − ϕ n +1 ) = (1 + b ω ∗ )(1 − ϕ n +1 ) = n + 1 . Hereby we mean ω = ∞ iff ϕ = − n and likewise for other identities. See [6, Corollary 8.5]for a generalization. In terms of ϕ j , ϕ j Khintchine’s estimates (3) simply read(11) ϕ n +1 ≥ − ϕ n , ϕ ≤ − ϕ n +1 n . The deduction from (T1), (T2) is carried out in Remark (b) in [21]. Equivalent formula-tions of (4) obtained similarly can be found in [22, (1.20), (1.20 ′ )]. Moreover (T1), (T2)imply that (BL1), (BL2) are respectively equivalent to(SS1) nϕ + ϕ n +1 ≤ − ϕ n +1 · (cid:18) n + 1 n − ϕ + 2 n − ϕ n +1 (cid:19) and(SS2) nϕ n +1 + ϕ ≥ − ϕ · (cid:18) n + 1 n − ϕ n +1 + 2 n − ϕ (cid:19) . This was again already observed by Schmidt and Summerer [22] who provided inde-pendent proofs of (SS1), (SS2) based on parametric geometry of numbers, and therebynew proofs (BL1), (BL2). Other proofs of (BL1), (BL2) that again rely on parametricgeometry of numbers came as a byproduct in [19], see [19, (23)] and [19, Remark 6].The latter proofs from [19] give some information on ( ϕ ( q ) , . . . , ϕ n +1 ( q )) as a function[0 , ∞ ) → [ − n, n +1 , i.e. on the combined graph defined in Section 3.2 below, in case ofequality. This observation inspired this note.In recent years, much work has been done on the joint spectrum of exponents, that isthe subset of R n +2 that occurs as( ϕ , . . . , ϕ n +1 , ϕ , . . . , ϕ n +1 ) ∈ R n +2 when ξ ∈ R n runs through all vectors that are Q -linearly independent together with { } .See for example [14], [15], [17], [18], [22], [24], [25], in particular the detailed expositionin Section 1 of [18]. In view of the equivalent claims (SS1), (SS2), our claims in Section 2contribute to this area by providing new results on the projection to the three-dimensionalspaces with coordinate variables ( ϕ , ϕ n +1 , ϕ n +1 ) and ( ϕ , ϕ , ϕ n +1 ), respectively. SeeTheorems 4.1, 4.2 below. We remark that for n = 2, a complete description of the joint JOHANNES SCHLEISCHITZ spectrum via a system of inequalities was given in [18, Theorem 11.5], thereby containingimplicitly the work of Laurent [10], Schmidt, Summerer [25] as well as the case n = 2 ofour Theorems 2.1, 2.2. However, the explicit deductions from [18] seem cumbersome.3.2. n -templates and the variational principle. From the functions ϕ j ( q ) and λ j ( q )associated to ξ ∈ R n as defined in Section 3.1, we derive L j ( q ) = qϕ j ( q ) = log λ j ( q ).These functions are piecewise linear with slopes among {− n, } , and by Minkowski’sSecond Convex Body Theorem their sum is uniformly bounded(12) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) n +1 X j =1 L j ( q ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ C ( n ) , q ∈ [0 , ∞ ) . The values ϕ j , ϕ j are just the extremal average slopes of the L j in a start segment, i.e.(13) ϕ j = lim inf q →∞ L j ( q ) q , ϕ j = lim sup q →∞ L j ( q ) q , (1 ≤ j ≤ n + 1) . We call L ξ ( q ) = ( L ( q ) , . . . , L n +1 ( q )) on q ∈ [0 , ∞ ) the combined graph associated to ξ ∈ R n . We approximate it by easier systems P = ( P ( q ) , . . . , P n +1 ( q )) without errorterm as in (12), but where we may glue consecutive functions P j on intervals where theydiffer by a small amount. Let Z ( j ) = { j, j − − n } , P ( q ) = −∞ , P n +2 ( q ) = + ∞ . Then following [4, Definition 5.1], an elegant formal description of the family of functionswe consider can be stated as follows.
Definition 1.
We call a continuous, piecewise-linear map P : [0 , ∞ ) → R n +1 an n -template if the component functions P j ( q ) satisy ( i ) − ( iv ) below.( i ) We have(14) n +1 X j =1 P j ( q ) = 0 , q ∈ [0 , ∞ ) . ( ii ) P ( q ) ≤ P ( q ) ≤ · · · ≤ P n +1 ( q ) for all q ∈ [0 , ∞ )( iii ) For any q ∈ [0 , ∞ ) and 1 ≤ j ≤ n + 1, if P j is differentiable at q then − n ≤ P ′ j ( q ) ≤ . ( iv ) For j = 0 , , . . . , n + 1 and every interval where P j < P j +1 holds, the function F j ( q ) = P ( q ) + P ( q ) + · · · + P j ( q )is convex and has slopes F ′ j in Z ( j ).In place of ( i ) it would suffice to demand P (0) = 0. Indeed P ( q ) is an n -templateiff P ( q ) /n is a balanced n × P j is one-sided differentiable on q ∈ [0 , ∞ ) with slopes within the finite set { } ∪ {− k/ ( n + 1 − k ) : 0 ≤ k ≤ n } . Wefurther remark that generalizing 3-systems defined by Schmidt, Summerer [25] naturallyto ( n + 1)-systems for n ≥
1, leads to special cases of n -templates, with slopes of the PTIMALITY OF TWO INEQUALITIES 7 P j restricted to {− n, } as for L j . More precisely, equipping the space of functions[0 , ∞ ) → R n +1 with the supremum norm, the closure of the set of ( n + 1)-systems in [25]becomes our set of n -templates. By ( i ), the sum of the slopes at points of differentiabilityvanishes as well. In the special case of ( n + 1)-systems this means one component decayswith slope − n while the remaining P j rise with slope +1. We further want to notice thatupon applying some affine map, our n -templates correspond to the generalized ( n + 1)-systems defined by Roy [17, Definition 5.1]. So an equivalent formulation of n -templatescan be derived from a twist of [17, Definition 4.1]. This also implies that the resultsin [16], [17], [18], remain basically valid and we will use them occasionally below.As noticed in [4], the following special case of [4, Theorem 5.2] is already covered bySchmidt and Summerer [22] and Roy [17, Corollary 4.7]. Theorem 3.1.
For any ξ ∈ R n , its associated combined graph has uniformly boundeddistance from some suitable n -template P , i.e. (15) k L ξ ( q ) − P ( q ) k = max ≤ j ≤ n +1 | P j ( q ) − L j ( q ) | ≤ C ( n ) , q ∈ [0 , ∞ ) . Conversely, for any n -template P , there is ξ ∈ R n inducing (15) for some effective C ( n ) . In particular, if P is as in the theorem for ξ , then by (13) we see(16) lim sup q →∞ P j ( q ) q = ϕ j , lim inf q →∞ P j ( q ) q = ϕ j . Theorem 3.2 below refines the latter claim of Theorem 3.1 by adding metrical informa-tion. For P an n -template, define its local contraction rate at q ∈ [ q , ∞ ] by δ ( P , q ) = κ − , κ := { max j : P ′ j ( q ) < } . This agrees with the definition of δ ( f , I ) in [4, (5.10)] for n × n -templates. See also [11]. Derive the average contraction rate in the interval [ q , q ] by∆( P , q ) = 1 q − q · Z qq δ ( P , u ) du and the upper and lower contraction rates by δ ( P ) = lim inf q →∞ ∆( P , q ) , δ ( P ) = lim sup q →∞ ∆( P , q ) . Denote by dim H resp. dim P the Hausdorff resp. packing dimensions, and call a family F of templates closed under finite perturbation if when P , Q are n -templates and P ∈ F and k P − Q k < ∞ , then also Q ∈ F . Then a consequence of the variational principle [4,Theorem 5.3] can be stated as follows. Theorem 3.2.
Let F be a family of n -templates closed under finite perturbation. Denoteby M = M ( F ) the set of all ξ ∈ R n whose associated combined graphs L ξ ( q ) satisfy k L ξ ( q ) − P ( q ) k < ∞ for some (thus all) P ∈ F . Then dim H ( M ) = sup P ∈ F δ ( P ) , dim P ( M ) = sup P ∈ F δ ( P ) . JOHANNES SCHLEISCHITZ
In particular, for any given n -template P , taking F the ”finite perturbation hull” of { P } and deriving M ( F ) = M ( { P } ) as above, we seedim H ( M ) ≥ δ ( P ) , dim P ( M ) ≥ δ ( P ) . As remarked in [3], there is no equality in general as δ is sensitive to perturbations.4. Reformulating claims in terms of ϕ j , ϕ j Equivalent formulations of Theorems 2.1, 2.2.
For n ≥ x ∈ [ − n,
1] write(17) g n ( x ) = (3 − n ) x + 1 − n + p (1 − x )[(4 n − x + 4 n − n + 1]2( n − . An equivalent formulation of Theorem 2.1 in terms of the quantities ϕ j , ϕ j that we willprefer to prove reads as follows. Theorem 4.1.
Let n ≥ be an integer. Then precisely for triples ( t, µ, σ ) satisfying (18) t ∈ [0 , , µ ∈ [ g n ( t ) , − tn ] , σ = (1 − n ) t + nµn + 1 + 2 t + ( n − µ there is ξ ∈ R n inducing equality in (SS1) and with associated values ϕ n +1 = t, ϕ n +1 = σ, ϕ = µ. For given ( t, µ, σ ) obeying the restrictions (18) , the set of ξ with these properties (19) Θ = Θ ( n ) t,µ = { ξ ∈ R n : ϕ n +1 = t, ϕ n +1 = σ, ϕ = µ } in fact has Hausdorff dimension at least n − and packing dimension at least n − /n . Write µ := g n ( t ) in the sequel. The deduction of Theorem 2.1 from Theorem 4.1relies purely on (T1), (T2). By equivalence of (3) and (11) the upper bound − t/n is aconsequence of the bound ρ and again the best we can hope for. For ρ , we further usethe defining equation(20) (1 + t + ( n − µ ) − (1 − t )(1 − µ ) = 0of µ . Moreover, by (T1), (T2) we may write t = 1 − n + 11 + ω ∗ = ω ∗ − n ω ∗ , µ = n + 1 ω + 1 − n = 1 − nωω + 1 . Inserting for t and µ = µ in (20) leads to a quadratic equation for ω in ω ∗ with (thecorrect) solution ρ , we omit the elementary calculation. Clearly, the argument can beread in reverse direction and the claims are indeed equivalent. We remark that if µ = µ ,as ϕ n +1 → ϕ n +1 → ϕ → − / ( n − ϕ → − /n .We refine the metrical claim of Theorem 4.1 in Section 4.2.Keep g n defined in (17). An equivalent formulation of Theorem 2.2 is the following. PTIMALITY OF TWO INEQUALITIES 9
Theorem 4.2.
Let n ≥ be an integer. Then precisely for triples ( s, ν, γ ) satisfying (21) s ∈ [ − n, , ν ∈ [ − sn , g n ( s )] , γ = (1 − n ) s + nνn + 1 + 2 s + ( n − ν there is ξ ∈ R n inducing equality in (SS2) and with associated values ϕ = s, ϕ = γ, ϕ n +1 = ν. For given ( s, ν, γ ) obeying the restrictions (21) , the corresponding set (22) Σ ( n ) s,ν = { ξ ∈ R n : ϕ = s, ϕ = γ, ϕ n +1 = ν } has packing dimension at least / , and positive Hausdorff dimension as soon as s > − n . The equivalence is derived similarly as for Theorem 4.1, we omit details. If ν = ν := g n ( s ), as ϕ → − n we calculate ϕ → − ( n − / ϕ n +1 →
1, in fact ϕ n +1 → Refinements of the metrical claims.
For simplicity we state our metrical claimsin the language of Section 4.1 only. Corresponding results in terms of classical exponentscan be inferred from (T1), (T2), which in particular imply the equivalences t → + ⇐⇒ ω ∗ → n + , t → − ⇐⇒ ω ∗ → ∞ , µ = − tn ⇐⇒ ω = ρ , µ = µ ⇐⇒ ω = ρ , and s → − ⇔ ω → n + , s → − n + ⇔ ω → ∞ , ν = − sn ⇔ ω ∗ = τ , ν = ν ⇔ ω ∗ = τ . We start with refining Theorem 4.1. Let A = A ( n ) t,µ = 1 − t t + ( n − µ · n + 1 · (cid:18) t + 2( n − µ − n + n (1 + t + ( n − µ ) − t (cid:19) ≥ , with A > t <
1, and derive B = B ( n ) t,µ and C = C ( n ) t,µ as(23) B = n − (2 − A )( n + 1) n + 1 + 2 t + ( n − µ , C = n − A n + 1 n + 1 + ( n − t + n ( n − µ . Then A ∈ [0 ,
2] and n ≥ B ≥ n − A, n ≥ C ≥ n − A for every t, µ as in Theorem 4.1.Recall the Hausdorff dimension of a set never exceeds its packing dimension [5]. Theorem 4.3.
The dimensions of the sets
Θ = Θ ( n ) t,µ in (19) are bounded from below by (24) dim H (Θ) ≥ n − A, dim P (Θ) ≥ max { B, C } . There is equality (at least) if t > and µ = µ . Moreover, max { B, C } ≥ n − n . We conjecture equalities in (24) as soon as t > µ ∈ [ µ , − t/n ), but only rigorouslyprove it for µ = µ in the last paragraph of Section 5.8. The restriction t > µ = − t/n etiher, as we explain below. We discuss special cases. If µ attains its maximum value µ = − t/n , then A ( n ) t, − t/n decreasesfrom 1 + ( n + 1) − to 1 as t rises in (0 , t → + dim P Θ ( n ) t, − t/n ≥ lim t → + dim H Θ ( n ) t, − t/n ≥ n − n + 1 , lim t → − dim H Θ ( n ) t, − t/n ≥ n − . As t → + then B t, − t/n , C t, − t/n tend to n − / ( n + 1) as well, suggesting equality inall estimates. We can pass to t = 0 , t = 1 by considering limiting graphs, so in particular(25) dim P (Θ ( n ) t, − t/n ) ≥ dim H (Θ ( n ) t, − t/n ) ≥ n − , t ∈ [0 , , The lower limit for the Hausdorff dimension is sharp, see the last paragraph of thissection. If µ = µ , evaluating the limits of A ( n ) t,µ , B ( n ) t,µ , C ( n ) t,µ as t → + and t → − fromTheorem 4.3 we getlim t → + dim P (Θ ( n ) t,µ ) = lim t → + dim H (Θ ( n ) t,µ ) = n − n + 1 , and(26) lim t → − dim H (Θ ( n ) t,µ ) = n − , lim t → − dim P (Θ ( n ) t,µ ) = n − n . On the other hand, if t = 0 equivalent to ϕ j = ϕ j = 0 for all 1 ≤ j ≤ n + 1, we have(27) dim P (Θ ( n )0 , ) = dim H (Θ ( n )0 , ) = n, since then ω ∗ = n , ω = 1 /n , which holds for almost all ξ ∈ R n . More generally, we expectthe maps ( t, µ ) dim H (Θ ( n ) t,µ ) and ( t, µ ) dim P (Θ ( n ) t,µ ) to be discontinuous on thecurve µ = − t/n where σ = 0. In this case our constructions in the proof below can berefined to obtain larger dimensions than in Theorem 4.3.We turn towards Theorem 4.2. Let(28) D = D ( n ) s,ν := n − s − s s + ( n − ν ≥ D > s > − n , and derive E = E ( n ) s,ν , F = F ( n ) s,ν via(29) E = n − ( n − D ) ( n + 1)( s + ( n − ν + 1)(1 − s )( n + 1 + s − ν ) , F = D n + 1 n + 1 + 2 s + ( n − ν . Then D ∈ [0 , n ] and n ≥ E ≥ D, n ≥ F ≥ D for all s, ν in the parameter range. Theorem 4.4.
The dimensions of the sets
Σ = Σ ( n ) s,ν in (22) are bounded by (30) dim H (Σ ( n ) s,ν ) ≥ D, dim P (Σ ( n ) s,ν ) ≥ max { E, F } . There is equality (at least) if s < and ν = ν . Moreover, max { E, F } ≥ . We again conjecture equalities if s < ν ∈ ( − s/n, ν ], but can guarantee it onlyfor ν = ν , see Section 6.3. Again s < ν = − s/n are vital. If ν = ν then as s → D, E, F become n − / ( n + 1) so we getlim s → − dim H (Σ ( n ) s,ν ) = lim s → − dim P (Σ ( n ) s,ν ) = n − n + 1 . PTIMALITY OF TWO INEQUALITIES 11 As s increases in [ − n, D ( n ) s,ν for the Hausdorff dimensions decays whereasmax { E ( n ) s,ν , F ( n ) s,ν } increases. In the other extremal case ν = − s/n , the values D ( n ) s, − s/n arestrictly increasing with s and evaluating limits we getlim s → − dim H (Σ ( n ) s, − s/n ) ≥ lim s → − D ( n ) s, − s/n = lim s → − E ( n ) s, − s/n = lim s → − F ( n ) s, − s/n = n − n + 1 , with right hand side in (0 , s → − n , we explain below that we must havelim s →− n + dim H (Σ ( n ) s,ν ) = lim s →− n + D ( n ) s,ν = 0 , ν ∈ [ − s/n, ν ] , (31)but we verify strictly positive dimensions as soon as s < − n . Furthermorelim s →− n + dim P (Σ ( n ) s,ν ) = lim s →− n + F ( n ) s,ν = 12 , and since D, E, F decrease in s, ν , we infer the lower bound 1 / ν = − s/n the limits as s → − n of both E, F become n − / ( n + 1), solim s →− n + dim P (Σ ( n ) s, − s/n ) ≥ n − n + 1 . Moreover, as soon as s > − n , the bounds for the dimensions become positive, for every ν ∈ [ − s/n, ν ]. Again we havedim H (Σ ( n )0 , ) = dim P (Σ ( n )0 , ) = n and we expect the maps ( s, ν ) dim H (Σ ( n ) s,ν ) and ( s, ν ) dim P (Σ ( n ) s,ν ) to be discon-tinuous on the curve ν = − s/n where γ = 0.We justify (31). If s → − n then ω → ∞ by (T1), and the sets W ( n ) w := { ξ ∈ R n : ω ( ξ ) ≥ w } have Hausdorff dimension ( n + 1) / ( w + 1) (see Jarn´ık [7]) which tendsto 0 as w → ∞ . On the other hand, t → ω ∗ → ∞ and the according sets V ( n ) w ∗ := { ξ ∈ R n : ω ∗ ( ξ ) ≥ w ∗ } have Hausdorff dimension at least n − w ∗ = ∞ , see [1]. This shows the bound in (25) is optimal and explains the dimension dropfrom Theorem 4.3 to Theorem 4.4. Our results complement metrical findings in [3], [4].For example the set of ”dually infinite singular vectors” S ( n ) ∞ ∗ := { ξ ∈ R n : b ω ∗ ( ξ ) = ∞} satisfies dim H ( S ( n ) ∞ ∗ ) = n − , dim P ( S ( n ) ∞ ∗ ) = n −
1, see [3, Section 1.2]. Since t → µ = µ imply b ω ∗ → ∞ by a comment below Theorem 2.1, in this sense the limitingHausdorff dimension in (26) is as large as possible, whereas the packing limit is not.5. Proof of Theorem 2.1
We have seen that Theorem 2.1 is equivalent to Theorem 4.1. We split the proof of thelatter in existence and non-existence part.
Generalized existence result.
For the existence part, we show the following moregeneral claim in the course of Sections 5.1-5.6 that also includes Theorem 4.3.
Theorem 5.1.
Let n ≥ and t ∈ [0 , . Derive (32) µ = µ ( n, t ) = (3 − n ) t + 1 − n + p (1 − t )[(4 n − t + 4 n − n + 1]2( n − . Then µ ≤ − t/n and for every µ ∈ [ µ , − t/n ] there exists a non-empty set Θ ∗ = Θ ∗ ( n ) t,µ ⊆ R n consisting of ξ = ξ t,µ ∈ R n and (33) ϕ n +1 = t, ϕ n +1 = ϕ n = (1 − n ) t + nµn + 1 + 2 t + ( n − µ , ϕ = · · · = ϕ n − = µ, and (34) ϕ = · · · = ϕ n − = nµ + tn + 1 + t − µ , and (35) ϕ n = − n + n + 1 n · − t n + ( n − t + n ( n − µ hold. In particular, (33) implies equality in (SS1) for any ξ ∈ Θ ∗ . The dimensions of Θ ∗ ( n ) t,µ satisfy the lower bounds (24) , with equality if t > and µ = µ . If µ = µ then the values in (34), (35) coincide, thus ϕ = · · · = ϕ n − = ϕ n . Moreover, ϕ n +1 = t and ϕ = µ and equality in (SS1) directly imply all claims (33), (34), (35), inparticular by (T1) we may express b ω as remarked in Section 2. Due to the additionalconditions (34), (35) we see Θ ∗ ( n ) t,µ ⊆ Θ ( n ) t,µ with Θ ( n ) t,µ from Theorem 4.1, so the latter andTheorem 4.3 are indeed implied. We will make use of the following calculations. Proposition 1.
Let n ≥ be an integer. For any t ∈ [0 , and µ = µ ( n, t ) defined in (32) we have − tn ≥ µ ≥ − t + (2 n + 1) tn − t ≥ − n − t. Proof.
We check the most challenging middle inequality first. Define F n ( x, y ) = (1 + x + ( n − y ) − ( x − y −
1) = 0 . We use that µ is solution to the quadratic equation(36) F n ( t, µ ) = (1 + t + ( n − µ ) − ( t − µ −
1) = 0 , equivalent to (20). Taking A n ( t ) = − ( t + (2 n + 1) t ) / ( n − t ) we see that for the resultingidentity G n ( t ) = F n ( t, A n ( t )) = 0then leads to a quartic polynomial with solutions t = 0 , t = 1 and a double solution t = − n / ∈ [0 , µ ( n, t ) , A n ( t ) and F n ( t ) in t we see that F n ( t, A n ( t )) and µ ( n, t ) − A n ( t ) do not change sign on [0 ,
1] (as otherwise µ ( n, t ) = A n ( t ) PTIMALITY OF TWO INEQUALITIES 13 for some t ∈ (0 ,
1) but then F n ( t, µ ) = F n ( t, A n ( t )) = G n ( t ) = 0, contradiction to G n nothaving zeros in (0 , µ ≥ A n ( t ) or µ ≤ A n ( t ) for all t in [0 , µ ≥ A n ( t ), are equivalent to H n ( t ) = (3 − n ) t + 1 − n + p (1 − t )[(4 n − t + 4 n − n + 1]2( n − + t + (2 n + 1) tn − t ≥ . Then we calculate H n (0) = H n (1) = 0 and H ′ n ( t ) = 2 n + 1 + 2 tn − t + ( t + (2 n − t )( n − t ) + 32( n − − n ( n − − n − n + 3 + (4 n − t n − p (1 − t )[(4 n − t + (2 n − ]and it is easily seen that H ′ n has a pole at t = 1 and the limit is −∞ as t approaches 1from below. Thus H n ( t ) and hence µ ( n, t ) − A n ( t ) are non-negative for t ∈ [1 − ǫ, t ∈ [0 , B n ( t ) = − t/n insteadof A n ( t ). Then I n ( t ) = F n ( t, B n ( t )) = 0 has solutions − n, t and a similar argumentshows µ ( n, t ) − B n ( t ) does not change sign on [0 , t ∈ [0 , (cid:3) By Theorem 3.1, it suffices to construct for given t ∈ [0 , , µ ∈ [ µ , − t/n ] an n -template P = P t,µ inducing the upper and lower limit values ϕ j , ϕ j as in Theorem 5.1. We mayassume 0 < t < ϕ , . . . , ϕ n +1 ) ⊆ R n +2 settled in [18] and the continuous dependency of the boundsfor σ, µ from t in the theorem. The case t = 0 implies ϕ j = ϕ j = 0 for all j , whichyields the trivial n -template P j ( q ) = 0 for 1 ≤ j ≤ n + 1, q ∈ [0 , ∞ ) anyway. It satisfiesTheorem 5.1 and puts the origin in the spectrum of ( ϕ , . . . , ϕ n +1 ) ∈ R n +2 . We shouldnote that Roy’s spectrum differs from ours, but it is easily seen that compactness ispreserved when switching between the formalisms. In the sequel, we call q a switch pointof P if some P j is not differentiable at q , i.e. P j has a local maximum or minimum.5.2. Preperiod of P t,µ . As customary when applying the variational principle, we wantto define an n -template with a periodic pattern. First we describe how to obtain theinitial state of the repeating construction. For given t ∈ (0 ,
1) and µ ∈ [ µ , − t/n ] with µ as in (32), the goal is the following scenario: At some q > P ( q ) q = · · · = P n − ( q ) q = µ, P n ( q ) q = θ, P n +1 ( q ) q = t, which is the starting point of Figure 1 below. Here according to (14) we put(38) θ = − ( t + ( n − µ ) . Moreover, at q we want the function P n +1 to decay with slope − n and P , . . . , P n risewith slope +1. It follows easily from the prescribed range for µ and Proposition 1 that(39) − n ≤ − n − ≤ µ ≤ µ ≤ − tn ≤ θ ≤ t ≤ , t ∈ [0 , , so indeed the values P j ( q ) /q belong to the required interval [ − n, q >
0. We startat q = 0 with P (0) = . . . = P n (0) = 0. Let P n +1 rise with slope +1 until some switchpoint q ′ ∈ (0 , q ], and then decay with slope − n until q , where q ′ is chosen so that P n +1 ( q ) = tq . By equating P n +1 ( q ) = tq = q ′ − n ( q − q ′ ) , we see q ′ = n + tn + 1 q . To get the desired value for P n ( q ), we let P n together with P , . . . , P n − initially decaywith slope − /n up to some switch point q ′′ where P n starts rising with slope 1 up to q ,with q ′′ chosen so that P n ( q ) = θq . One determines q ′′ = (1 − θ ) nn + 1 q ≤ q ′ , where the inequality holds due to (39). We remark that q ′′ = 0 if t = 1. We take theremaining first n − P , . . . , P n − identical in [0 , q ], so thatthey are determined in [0 , q ] by the vanishing sum property (14) and the descriptionof P n , P n +1 above. This means P , . . . , P n − decay with slope − /n in [0 , q ′′ ], with slope − / ( n −
1) in [ q ′′ , q ′ ] and finally rise with slope +1 in [ q ′ , q ]. This concludes the preperiod.5.3. Period of P t,µ : Special case µ = µ . We take µ = µ throughout this sectionand consequently impliclty consider θ = θ derived from (38) with this choice. We remarkthat by our choice of µ , we verify that t, µ = µ , θ = θ in (38) are linked by the quadraticidentity(40) θ − θ − µt + t + µ = ( θ − − ( µ − t −
1) = 0 , which reflects (20).The figure shows the first period [ q , q ] of the iterative construction, starting from q where (37) holds, up to some q > q to be determined below. We continue the slopes asfrom the preperiod at q to the right, i.e. slope − n for P n +1 and +1 for P , . . . , P n . Atsome point ˜ q > q the functions P n , P n +1 will intersect. By equating P n +1 (˜ q ) = tq − n (˜ q − q ) = θq + (˜ q − q ) = P n (˜ q ) , and a brief computation, upon inserting (38), this point is given by(41) ˜ q = n + 1 + 2 t + ( n − µn + 1 · q PTIMALITY OF TWO INEQUALITIES 15 and induces the quotients(42) P n (˜ q )˜ q = P n +1 (˜ q )˜ q = (1 − n ) t + nµn + 1 + 2 t + ( n − µ . Note that the right hand side of (42) is the desired slope for ϕ n +1 and ϕ n in (33). At˜ q the functions P n , P n +1 exchange slopes, so P n starts decaying with slope − n and P n +1 rising with slope +1. Then at some point ˜ q > ˜ q the rising functions P = · · · = P n − will intersect P n . From(43) P n (˜ q ) = tq − n (˜ q − q ) = µq + (˜ q − q ) = P (˜ q ) · · · = P n − (˜ q )we derive(44) ˜ q = t − µ + n + 1 n + 1 q . To the right of this switch point ˜ q , let the slopes of P , . . . , P n − become − / ( n − P n becomes +1. Observe P n +1 also still increases with slope +1.Since we can assume t <
1, then at some point q ≥ ˜ q we will have P n +1 ( q ) /q = t ,concretely from imposing P n +1 ( q ) = θq + ( q − q ) = q t we calculate(45) q = θ − t − q = µ − θ − q , where the right equality uses µ = µ , θ = θ and reflects (40). With a little effort it canbe checked, and follows from the below calculations, that actually q ≥ ˜ q . Indeed bysome rearrangements this is equivalent to t + (2 n + 1 − µ ) t + n µ ≥ ⇐⇒ µ ≥ − t + (2 n + 1) tn − t , which is confirmed in Proposition 1. Moreover, at the same point q we further have P n ( q ) q = µq + ( q − q ) q = θ, P ( q ) q = · · · = P n − ( q ) q = µ. The left identity is equivalent to (45), the right follows consequently from P n +1 ( q ) /q = t ,the vanishing sum property (14) and P ( q ) = · · · = P n − ( q ). Finally we let q be a switchpoint where P n +1 starts decaying with slope − n , whereas P , . . . , P n − start rising withslope +1. Note P n is differentiable at q with slope +1 and clearly q > q if t >
0, whichwe can assume.Since the slopes as well as the right-sided derivatives of all P j at q are then identicalto the data at q , at q we have precisely the same conditions as at q . Thus, up to ascaling factor q /q in each step, we can extend the construction from [ q , q ] periodicallyad infinitum. Together with the preperiod, this will give an n -template on [0 , ∞ ). qP(q) ϕ n +1 = tθ ϕ = · · · = ϕ n − = µ ϕ n +1 = ϕ n = σϕ = · · · = ϕ n − = ϕ n P n +1 P = · · · = P n − P n P n +1 P = · · · = P n − q ˜ q ˜ q q P n +1+1 − n − +1 − n Figure 1: Sketch period of P t,µ (special case µ = µ )5.4. Period of P t,µ : The general case.
Now let µ be arbitrary in [ µ , − t/n ]. We startthe period construction as in the special case µ = µ : After the preperiod we derive atsome q > θ = − ( t + ( n − µ ) we are given P ( q ) q = · · · = P n − ( q ) q = µ, P n ( q ) q = θ, P n +1 ( q ) q = t. We describe the period [ q , q ]. Starting from q , we let P n +1 decay with slope − n andthe others rise with slope +1 until at ˜ q the functions P n , P n +1 meet and exchange slopes.Then at some point ˜ q > ˜ q the functions P = · · · = P n − meet P n . The calculationsfor ˜ q , ˜ q and P n (˜ q ) are precisely as in the special case in Section 5.3, for general µ andimplied θ throughout.Now at ˜ q we make a twist to our construction above by prescribing that all P , . . . , P n decay with slope − /n in [˜ q , ˜ q ] for some ˜ q ≥ ˜ q to be determined, while P n +1 keepsincreasing with slope +1. At some point q > q we will have P n +1 ( q ) /q = t again. Itis obvious that q > ˜ q and since µ ≥ µ it follows from the argument in the special caseand µ ≥ µ that even q ≥ ˜ q . The identity (42) and the left identity in (45) hold for thesame reason as in the special case, in particular q is evaluated as for µ = µ .We will choose ˜ q ≤ q and in the interval [˜ q , q ] we let P n rise with slope +1 and P , . . . , P n − decay with slope − / ( n − q = ˜ q then the graph is as PTIMALITY OF TWO INEQUALITIES 17 for µ = µ ). We claim that upon a proper choice of ˜ q ∈ [˜ q , q ], at q again we will have P ( q ) q = · · · = P n − ( q ) q = µ, P n ( q ) q = θ, P n +1 ( q ) q = t. Assume this is shown. Then the period is finished and again we repeat the constructionupon scaling by q /q in each step.To evaluate ˜ q , we have to satisfy the identity θq = P n (˜ q ) − n (˜ q − ˜ q ) + ( q − ˜ q ) , thus inserting for θ we get˜ q = nn + 1 (cid:20) P n (˜ q ) + 1 n ˜ q + (1 + t + ( n − µ ) q (cid:21) . Inserting from (43), (44), (45) we get˜ q = nn + 1 (cid:20) t + n + 1 − nn ( t − µ + n + 1) + (1 + t + ( n − µ ) − t (cid:21) · q = 1 + t + ( n − µn + 1 · n + ( n − t + nµ )1 − t · q . (46)We now finally check analytically that indeed ˜ q ∈ [˜ q , q ] as claimed for any choice n ≥ t ∈ (0 ,
1) and µ ∈ [ µ , − t/n ].Writing q /q = (1 − θ ) / (1 − t ) and ˜ q /q = (1 − θ )(1 + n (1 − θ ) / (1 − t )) / ( n + 1), theclaim ˜ q ≤ q can be rearranged to θ ≥ − t/n which is true by (39). For the estimate˜ q ≤ ˜ q , we check by inserting that there is equality ˜ q = ˜ q at the minimum value µ = µ ,which agrees with the special case from the section above. Since ˜ q decreases with slope − / ( n + 1) as a function of µ by (44), to conclude it suffices to check that for fixed t thevalue ˜ q in (46) increases as a function of µ . We calculate ddµ ˜ q = ( n − n − t + 2 n + 1 + 2 n ( n − µ ]( n + 1)(1 − t ) q . Since t ∈ [0 , n − t + 2 n + 1 + 2 n ( n − µ ≥ . Using µ ≥ µ ≥ − n − t by Proposition 1 and t ∈ [0 ,
1] this can be readily verified. Thiscompletes the construction of first period [ q , q ] for general µ . The periodic continuationto an n -template on [0 , ∞ ) via adjacent scaled copies is performed as in Section 5.3. qP(q) ϕ n +1 = tθϕ n +1 = ϕ n = σϕ = · · · = ϕ n − = µϕ = · · · = ϕ n − ϕ n P n +1 P = · · · = P n P n P n +1 P = · · · = P n − P = · · · = P n − q ˜ q ˜ q ˜ q q P n +1 +1 − n − n − +1 − n Figure 2: Sketch period of P t,µ (general case)5.5. Evaluation of extremal values.
Since the maximum slope P n +1 ( q ) /q of P n +1 for q ∈ [ q , q ] is clearly attained at the interval ends where it takes the value t , and similarlythe minimal slopes for P = · · · = P n − at the interval ends equal to µ , we see thatlim inf q →∞ P j ( q ) q = µ, (1 ≤ j ≤ n − , lim sup q →∞ P n +1 ( q ) q = t. Moreover, obviously the expression P n +1 ( q ) /q for q ∈ [ q , q ] is minimal at its local min-imum ˜ q , and similarly the slope of P n attains its maximum within [ q , q ] at ˜ q . FromTheorem 3.1 and (13), (16), (42) we conclude (33).Finally, for (34), (35) we show that within q ∈ [ q , q ], the values P ( q ) /q = · · · = P n − ( q ) /q take their maxima at ˜ q , and P n ( q ) /q its minimum at ˜ q . It is clear that theextrema in question are taken in the interval I = [˜ q , ˜ q ], since outside the slopes take theextremal values − n and 1. We show that P n (˜ q ) / ˜ q = ( nµ + t ) / ( n + 1 + t − µ ) exceedsthe slope − /n of P , . . . , P n in I . Then the values P j ( q ) /q for j = 1 , , . . . , n decreasewithin I and the claim follows. To show nµ + tn + 1 + t − µ > − n PTIMALITY OF TWO INEQUALITIES 19 we rearrange to the equivalent form n + 1 > ( n − t + ( n + 1) µ which is true since as t ∈ [0 ,
1] and µ ≤ n + 1 > n − ≥ ( n − t ≥ ( n − t + ( n + 1) µ . The claim (34) follows directly. For (35), from (43), (44), (46) we calculate ϕ n = P n (˜ q )˜ q = P n (˜ q ) − ˜ q − ˜ q n ˜ q = − n + n + 1 n · t + ( n − µ t + ( n − µ + n (1+ t +( n − µ ) − t . Dividing numerator and denominator by θ = 1 + t + ( n − µ yields the claimed expressionafter a brief rearrangement. We again conclude with Theorem 3.1 and (13), (16). Finally,inserting for ϕ , ϕ n +1 , ϕ n +1 from (33), a calculation verifies equality in (SS1).5.6. Deduction of metrical results.
We bound the Hausdorff and packing dimensionsof the set Θ ∗ ( n ) t,µ in Theorem 5.1 as in (24). First assume 0 < t < ∗ ( n ) t,µ is contained in Θ ( n ) t,µ from Theorem 4.1,clearly (24) follows. We determine the contraction rates for the n -template P = P t,µ constructed above. We evaluate the local contraction rates within the period interval[ q , q ] as(47) δ ( P , q ) = n, if q ∈ [ q , ˜ q ] ,n − , if q ∈ [˜ q , ˜ q ] ,n − , if q ∈ [˜ q , q ] . Denoting for j ≥ j -th period interval I j = [ q j − , q j ], this is true accordingly in I j .From the variational principle we directly conclude that the Hausdorff and packing di-mensions cannot be less than n −
2. For the precise calculation, we observe that the localrate decays within each interval I j . We readily conclude that the lower limit is attainedwhen considering intervals [ q , q N ], and in fact by periodicity the resulting average con-traction rate in these intervals is independent of N ≥
1. So from the variational principleTheorem 3.2 we getdim H (Θ ∗ ( n ) t,µ ) ≥ δ ( P ) = R q q δ ( P , q ) q − q = n (˜ q − q ) + ( n − q − ˜ q ) + ( n − q − ˜ q ) q − q = ( n − q + ˜ q + ˜ q − nq q − q = n − q + ˜ q − q q − q . Inserting for ˜ q , ˜ q , q from (41), (45), (46) we verify dim H (Θ ∗ ( n ) t,µ ) ≥ n − A as in (24).For the upper limit, we consider intervals [ q , ˜ q ,N ] and [ q , ˜ q ,N ] for large N , where ˜ q i,N denotes for i = 1 , , q i ∈ [ q , q ] in the interval [ q N − , q N ]. Inparticular q N ≤ ˜ q ,N ≤ ˜ q ,N ≤ q N +1 for all N and the contraction rates in the subintervals[ q N , ˜ q ,N ], [˜ q ,N , ˜ q ,N ] and [˜ q ,N , q N +1 ] take the values as for N = 0 in (47). Hencedim P (Θ t,µ ) ≥ δ ( P ) ≥ max { S, T } where S and T are respecitvely the average limit contraction rates in the intervals [ q , ˜ q N, ]and [ q , ˜ q N, ] respectively as N → ∞ . In fact it is not hard to check equality δ ( P ) =max { S, T } . To conclude, we show S ≥ B, T ≥ C with B, C as in (23). Since we identified δ ( P ) as the average contraction rate in any interval [ q , q N ] and q = o ( q N ) as N → ∞ ,we evaluate S = lim N →∞ R ˜ q ,N q δ ( P , q ) dq ˜ q ,N − q = lim N →∞ δ ( P )( q N − q ) + n (˜ q ,N − q N )˜ q ,N − q = lim N →∞ δ ( P ) q N + n (˜ q ,N − q N )˜ q ,N = n − ( n − δ ( P )) · lim N →∞ q N ˜ q ,N . Now since q N / ˜ q ,N is independent of N , inserting δ ( P ) ≥ n − A we infer S ≥ n − q ˜ q ( n − δ ( P )) ≥ n − (2 − A ) q ˜ q = n − (2 − A )( n + 1) n + 1 + 2 t + ( n − µ = B. For T a similar caclulation shows T = lim N →∞ R ˜ q ,N q δ ( P , q ) dq ˜ q ,N − q = lim N →∞ R q N +1 q δ ( P , q ) dq − R q N +1 ˜ q ,N δ ( P , q ) dq ˜ q ,N − q = lim N →∞ ( q N +1 − q ) δ ( P ) − ( n − q N +1 − ˜ q ,N )˜ q ,N − q = lim N →∞ q N +1 δ ( P ) − ( n − q N +1 − ˜ q ,N )˜ q ,N = n − δ ( P ) − n + 2) lim N →∞ q N +1 ˜ q ,N = n − δ ( P ) − n + 2) q ˜ q = n − A q ˜ q = n − A ( n + 1)(1 + t + ( n − µ )(1 − t )(1 + t + ( n − µ ) + n (1 + t + ( n − µ ) = n − A n + 1 n + 1 + ( n − t + n ( n − µ = C, as claimed, where we inserted for q , ˜ q from (45), (46) in the last line. Finally, for t = 0 the claim (24) is trivial by (27), and we can extend the formula to t = 1 byconsidering a limiting n -template, compare with [20, Section 2], we omit details. Theproof of Theorem 5.1 is complete.5.7. Extending the range of b ω ∗ . We sketch how to alter the graphs in Figure 2 to geta prescribed value for b ω ∗ as in the interval of (9). We keep q , ˜ q and the graph from P t,µ in [ q , ˜ q ] unchanged. We alter the formulas for ˜ q , ˜ q , q , still satisfying ˜ q ≤ ˜ q ≤ ˜ q ≤ q , and introduce a new point ˜ r between ˜ q and ˜ q . We let P n , P n +1 decay withslope − ( n − / q , ˜ r ] and then starting at ˜ r we let P n +1 rise with slope +1 and P n decay with slope − n until it meets P = · · · = P n − . The construction in [˜ q , q ] remains PTIMALITY OF TWO INEQUALITIES 21 basically as in P t,µ in Figure 2. For given η ∈ (cid:20) , (1 − n ) t + nµn + 1 + 2 t + ( n − µ (cid:21) , appropriate choices of ˜ r, ˜ q , ˜ q , q induce an n -template P t,µ,η satisfying (33) apart from ϕ n +1 altered to ϕ n +1 = η . Thus by (T2) we obtain any b ω ∗ as in (9) (remark: (34), (35)are not preserved). We omit the calculations and only want to illustrate qualitatively thegraph in Figure 3 below. We omit metrical claims derived from Theorem 3.2 as well.qP(q) ϕ n +1 = tθϕ n ϕ n +1 = ηϕ = · · · = ϕ n − = µϕ = · · · = ϕ n − ϕ n P n +1 P = · · · = P n − P = · · · = P n − P n q ˜ q ˜ q ˜ r ˜ q q − n +1+1 − n − − n +1 − n − +1 − n P n +1 Figure 3: Sketch period P t,µ,η (extended case)5.8. Non-existence in Theorem 4.1.
To establish the non-existence part of Theo-rem 4.1 means to show the following claim.
Theorem 5.2.
Let n ≥ , t ∈ [0 , and µ = µ ( n, t ) as in Theorem 4.1. Then for µ / ∈ [ µ , − t/n ] , the set Θ ( n ) t,µ of ξ ∈ R n that induces ϕ n +1 = t, ϕ = µ and equality in (SS1) is empty. In other words, no ξ ∈ R n induces (48) ϕ n +1 = t, ϕ n +1 = σ := (1 − n ) t + nµn + 1 + 2 t + ( n − µ , ϕ = µ. We prove the theorem. For µ > − t/n we cannot even have ϕ n +1 = t, ϕ = µ due tothe reverse inequality ϕ ≤ − ϕ n +1 /n in (11). It remains to contradict µ < µ upon theassumptions ϕ n +1 = t, ϕ = µ and equality in (SS1) of the theorem. Keep in mind for the sequel the equivalence in the claims of Theorem 5.2, i.e. upon ϕ n +1 = t, ϕ = µ , equalityin (SS1) is equivalent to ϕ n +1 taking the value σ in (48).Step 1: We show that equality in (SS1) implies that essentially the situation as in theinterval [ q , ˜ q ] ⊆ [ q , q ] in Figure 2 (or Figure 1) occurs for arbitrarily large q . For thiswe basically rephrase an argument within the proof of [19, Theorem 3.2]: Since ϕ n +1 = t ,for any ǫ > q with | L n +1 ( q ) − tq | ≤ ǫq . Choose large q with this property. For simplicity of notation, we omit ǫ and use o notation in the sequel,so we write L n +1 ( q ) = tq + o ( q ) and mean that in fact we consider a sequence of q values with this property that tends to infinity. We may assume that at q there is a localmaximum of L n +1 . Consider the next point q + ˜ q where L n , L n +1 meet to the right of q , i.e. ˜ q > L n ( q + ˜ q ) = L n +1 ( q + ˜ q ). By definition of ϕ n +1 clearly L n +1 ( q + ˜ q ) ≥ ( ϕ n +1 − ǫ )( q + ˜ q ) = ( σ − o (1))( q + ˜ q ) . Since L n has slope at most 1, we infer(49) L n ( q ) ≥ L n ( q + ˜ q ) − ˜ q = L n +1 ( q + ˜ q ) − ˜ q ≥ ( σ − q + σq − o ( q + ˜ q ) . Together with the bounded sum property (12), we infer(50) L ( q ) ≤ − L n ( q ) + L n +1 ( q ) n − O (1) ≤ − ( t + σ ) q + ( σ − qn − o ( q + ˜ q ) . We estimate ˜ q . Since L n +1 decays with slope − n in [ q , q + ˜ q ] and L n +1 ( q + ˜ q ) / ( q + ˜ q )is at least σ + o (1) by definition of ϕ n +1 = σ , we have L n +1 ( q + ˜ q ) = L n +1 ( q ) − n ˜ q ≥ ( σ − o (1))( q + ˜ q ) . Inserting L n +1 ( q ) = tq + o (1) q we get(51) ˜ q ≤ (cid:18) t − σn + σ + o (1) (cid:19) q . Since σ ≤
1, by (50) when dividing by q we get L ( q ) q ≤ − t + σ + ( σ − t − σn + σ n − o (1) . As we can assume ϕ ≤ L ( q ) /q + o (1), after some rearrangement when taking limitswe may drop the o (1) terms, and find the corresponding inequality ϕ ≤ − ϕ n +1 + ϕ n +1 + ( ϕ n +1 − ϕ n +1 − ϕ n +1 n + ϕ n +1 n − q as above tends to infinity, by(49), (51) we must have L n +1 ( q ) = tq + o ( q ) , L n ( q ) = (cid:18) ( σ − t − σn + σ + σ (cid:19) q + o ( q ) PTIMALITY OF TWO INEQUALITIES 23 and further from equality in (50) we infer L j ( q ) = − t + σ + ( σ − t − σn + σ n − q + o ( q ) , ≤ j ≤ n − . With some calculation, we check that when dropping the remainder terms, the expressionfor L n ( q ) /q agrees with the value θ from (38), and L ( q ) /q with µ . Upon identifying˜ q + q = ˜ q , this indeed verifies that essentially the combined graph in the interval [ q , ˜ q ]must look like in Figure 2 from the construction.Step 2: we show that if µ < µ we cannot extend the graph of Figure 2 from [ q , ˜ q ] tothe right of ˜ q without violating the requirements of a combined graph, thereby we geta contradiction. Let r > ˜ q be the first coordinate of the next meeting point of L n , L n +1 to the right of ˜ q , i.e. the smallest solution for L n ( r ) = L n +1 ( r ) with r > ˜ q . Write I = [˜ q , r ]. Now we distinguish two cases.Case 1: The functions L n − and L n do not meet in I , i.e. L n ( q ) > L n − ( q ) for q ∈ I .Then it is clear from the theory of combined graphs/ n -templates that, up to o ( q ), thegraph in I must look as follows: there is some switch point u ∈ I so that in [˜ q , u ] thefunction L n +1 must rise with slope 1 and decay in [ u, r ] with slope − n , whereas for some v = u + o ( u ) very close to u the opposite happens for L n , i.e. L n decays with slope − n in [˜ q , v ] and increases with slope +1 in [ v, r ]. (The functions L , . . . , L n − all rise withaverage slope +1 − o (1) in the entire interval I .) We justify this claim, but for brevityomit full rigorosity: First note that since L n − and L n do not meet in the interior of I ,any ”serious” local minimum of L n ( q ) at some q ∈ I induces a local maximum of L n +1 ( ℓ )at some ℓ = q + O (1). This can be seen by passing to a close n -template as in (15) andthe convexity condition in Definition 1 and (12). Now since L n and L n +1 move apart ina neighborhood to the right of ˜ q , the function L n must change slope to +1 somewherein I , and by the above argument in proximity L n +1 must change slope to − n . So clearlythere is at least one switch point in the interior of I where L n , L n +1 exchange slopes asabove. Assume there was another ”serious” switch point ˜ ℓ in the interior of I where L n changes slope. Then at ˜ ℓ , L n starts to decay with slope − n and L n +1 must start to risewith slope +1 by (12) and we assume these slopes continue to the right on a subintervalof I of substantial length. Since ˜ ℓ is in the interior of I , then in an associated close n -template satisfying (15), the function P n +1 would have a local minimum which is nota local maximum of P n . This contradicts the convexity condition of templates again.(Instead of passing to templates in the last step, we can alternatively argue with the firsttwo successive minima of the dual lattice point problem). This confirms our claim.In particular, the average slope of L n and L n +1 in I is − ( n − / o (1) <
0. Sinceclearly L n +1 ( q ) ≥ r we get L n +1 ( r ) /r < L n +1 (˜ q ) / ˜ q = ϕ n +1 + o (1),contradiction to ϕ n +1 ≤ L n +1 ( r ) /r − o (1) unless r − ˜ q is very small. However, it is clearthat we may assume this is not the case. For example, we may pass to n -templates again,or start with ε > r with r > (1 + ε )˜ q and then use the aboveargument. We omit the technical details.Case 2: The functions L n − and L n meet in I . Starting at its local minimum ˜ q whereit meets L n , the function L n +1 rises with slope +1. However, since ϕ n +1 = t , this happens at most until a point y on the first axis where L n +1 ( y ) /y = t + o (1). Then L n +1 decayswith slope − n until it meets L n at r > y > ˜ q . Since L n − and L n meet in I and no slopecan exceed +1, it is clear that(52) L n ( r ) ≤ L n − ( q ) + ( r − q ) . Recall that in the construction for µ = µ , the value y was as large as possible (up to o (1)) since L n +1 indeed went up until y with L ( y ) /y = t , and there was equality in (52),since at the meeting point ˜ q of L n − and L n the slope of L n changed from − n to +1and remained +1 until it met L n +1 . Further identifying our y with q from that proof,for µ = µ we had the minimum possible value L n +1 ( r ) /r = σ at r . So it is geometricallyobvious that if we start with µ < µ (which also implies larger values of σ and θ ), even inthe most disadvantageous case of maximal y and equality in (52), at the smallest point z > y where L n +1 ( z ) /z = σ , we will have L n +1 ( z ) ≥ (1 + ε ) L n ( z ) with some ε > µ, µ . We omit the explicit calculation. This means that r > (1 + ǫ ) z and L n +1 continues to decay with slope − n in [ z, r ] until it meets L n at r , for some ǫ >
0. Thus obviously ϕ n +1 ( r ) = L n +1 ( r ) /r ≤ (1 − ǫ ) L n +1 ( z ) /z = (1 − ǫ ) σ for some ǫ >
0. However, this contradicts the definition of σ = ϕ n +1 . This completes the proof ofTheorem 5.2.We finally observe that the equality in the dimension formulas (24) for t > µ = µ follows from the proof above. Our argument shows that then the combined graphmust indeed be composed from consecutive periods as in Figure 1, up to o ( q ) as q → ∞ .Take the family F of n -templates with these properties, which is closed under finiteperturbations in view of the error term. Then it is not hard to see that the supremaof δ ( Q ) , δ ( Q ) over Q ∈ F are attained for Q = P t,µ as constructed (since o ( q ) has anegligible effect in the limit and by changing slopes of some P j locally in intervals whereconsecutive functions P j are glued, we may only decrease the local contraction rate. Weskip details). Application of the variational principle yields the claim. Note that we losethe case t = 0, where (24) indeed fails as pointed out in Section 4.2, since then q = q inour construction by (45), so the period collapses to a singleton and we get no n -template.6. Proof of Theorem 2.2
By equivalence of Theorems 2.2, 4.2 we again may just prove Theorem 4.2, and weshow the following more general existence claim that includes Theorem 4.4.
Theorem 6.1.
Let n ≥ and s ∈ [ − n, . Derive ν = g n ( s ) with g n as in (17) . Then ν ≥ − s/n and for every ν ∈ [ − s/n, ν ] there exists a non-empty set Σ ∗ = Σ ∗ ( n ) s,ν consistingof ξ = ξ s,ν ∈ R n whose associated quantities ϕ j , ϕ j satisfy (53) ϕ = s, ϕ = ϕ = (1 − n ) s + nνn + 1 + 2 s + ( n − ν , ϕ = · · · = ϕ n +1 = ν, and ϕ = · · · = ϕ n +1 = nν + sn + 1 + s − ν , PTIMALITY OF TWO INEQUALITIES 25 and ϕ = − n + n + 1 n · − s n + ( n − s + n ( n − ν . Every ξ ∈ Σ ∗ ( n ) s,ν induces equality in (SS2) . The dimensions of Σ ∗ ( n ) s,ν are bounded as in (30) , with equality if s < and ν = ν . We again have Σ ∗ ( n ) s,ν ⊆ Σ ( n ) s,ν with Σ ( n ) s,ν from Theorem 4.2. Moreover, if ν = ν , then ϕ = ϕ = · · · = ϕ n +1 and additional equality in (SS2) induces all values ϕ j , ϕ j as in thetheorem. Again we construct suitable n -templates P s,ν in order to apply Theorem 3.1.The construction is dual in some sense. We omit certain computations that are similarto the proof of Theorem 5.1. We start with the dual version of Proposition 1. Proposition 2.
For any s, ν as in Theorem 6.1 we have − n − s ≥ ν ≥ − s + (2 n + 1) sn − s ≥ − sn . We skip the proof as it works very similar as in Proposition 1. We explain how weconstruct n -templates P s,ν with the desired properties. We may again assume strictinequalities − n < s < Preperiod of P s,ν . Similarly to Theorem 5.1, here for some q > P ( q ) q = · · · = P n +1 ( q ) q = ν, P ( q ) q = ϑ, P ( q ) q = s, where again ϑ is determined from s, ν in view of (14) via(55) ϑ = − ( s + ( n − ν ) . Moreover, at q the functions P , P rise with slope +1 while P , . . . , P n +1 decay withslope − / ( n − − n ≤ s ≤ ϑ ≤ − s/n ≤ ν ≤ ν ≤ s ∈ [ − n, s = − n is equivalent to ϑ = ν = 1.By choice of ϑ , for ν = ν = g n ( s ) we again have ϑ − ϑ − νs + ν + s = ( ϑ − − ( ν − s −
1) = 0 . To obtain (54), starting at q = 0 we let P decay with slope − n up to a switch point q ′ ∈ (0 , q ] where it starts increasing with slope +1 until q . Hereby q ′ is determined viathe property P ( q ) = sq , giving q ′ = ((1 − s ) / ( n + 1)) q . In [0 , q ′ ] we let all P , . . . , P n +1 rise with slope +1. At the switch point q ′ we start letting P decay with slope − n upto some point q ′′ ≥ q ′ while the other functions all rise with slope +1 in [ q ′ , q ′′ ]. Thenstarting from q ′′ we let P rise with slope +1 so that P , . . . , P n +1 have slopes − / ( n − q ′′ , q ]. A suitable choice of q ′′ will lead to P ( q ) /q = ϑ , and by P ( q ) /q = s , thevanishing sum property (14) and P ( q ) = · · · = P n +1 ( q ), actually all conditions in (54)are implied. Concretely q ′′ = ((2 − s − ϑ ) / ( n + 1)) q is derived from q ′ − n ( q ′′ − q ′ ) + ( q − q ′′ ) = ϑq and inserting for q ′ , and indeed q ′ ≤ q ′′ since this is equivalent to ϑ ≤ Period of P s,ν and conclusion.
It is convenient to give a reverse construction ofthe period, i.e. start from q > q where the properties(56) P ( q ) q = · · · = P n +1 ( q ) q = ν, P ( q ) q = ϑ, P ( q ) q = s, are satisfied and calculate back to derive the same conditions (54) at q , using our choiceof ϑ . It is clear that ultimately we can change the direction back to positive and repeatthe period [ q , q ], blown up by the constant factor q /q in each step, ad infinitum again.Again first consider the special case ν = ν = g n ( s ). Let q , q be related via(57) q = s − ϑ − q = ϑ − ν − q . The case ν = ϑ = 1 is equivalent to s = − n which we excluded. We determine ˜ q < q from intersecting the continuation of P to the left decreasing with slope − n with thelikewise continuation of P increasing with slope +1. From equating P and P at ˜ q weget P (˜ q ) = sq + n ( q − ˜ q ) = ϑq − ( q − ˜ q ) = P (˜ q ) . After some calculation and using (54), (55) we derive(58) ˜ q = n + 1 + 2 s + ( n − νn + 1 · q , P (˜ q )˜ q = P (˜ q )˜ q = (1 − n ) s + nνn + 1 + 2 s + ( n − ν , and we recognize the right hand side as the value from (53). When moving to the leftfrom ˜ q , we let P , P exchange slopes at ˜ q up to a point ˜ q < ˜ q where P intersects P = · · · = P n +1 that rise with slope +1 in [˜ q , q ]. From P (˜ q ) = · · · = P n +1 (˜ q ) = νq − ( q − ˜ q ) = sq + n ( q − ˜ q ) = P (˜ q )we calculate(59) ˜ q = n + 1 + s − νn + 1 q , P (˜ q )˜ q = nν + sn + 1 + s − ν . We further check by Proposition 2, and it follows from the construction below, that˜ q ≥ q . At the switch point ˜ q , when going to the left we change the slope of P to+1 and the slopes of P , . . . , P n +1 according to (14) to − / ( n − P still hasslope +1. Since P rises with slope +1 left of ˜ q , at some point q ∗ < ˜ q we will have P ( q ∗ ) = sq ∗ . We will check that q ∗ = q ≤ ˜ q and that keeping the slopes in [ q ∗ , ˜ q ]equations (54) hold. From equating P ( q ∗ ) = sq ∗ = ϑq − ( q − q ∗ )we indeed readily check that q ∗ = q is the value as in (57). Moreover, from (57) we verify P ( q ) = νq − ( q − q ) = ϑq . Since clearly P ( q ) = · · · = P n +1 ( q ) from (14) we conclude the remaining claims of (54),proving our assertion.Finally, for general ν ∈ [ − s/n, ν ], we again split the interval [ q , ˜ q ] into [ q , ˜ q ] and[˜ q , ˜ q ] for some q ≤ ˜ q ≤ ˜ q and let P , . . . , P n +1 all decay with slope − /n in [˜ q , ˜ q ],and in [ q , ˜ q ] we take the slopes − / ( n −
1) for P , . . . , P n +1 and +1 for P , i.e. as in PTIMALITY OF TWO INEQUALITIES 27 the interval [ q , ˜ q ] when ν = ν . The value ˜ q is again determined so that the imposedassumptions (54) at q are met. Similar to Theorem 5.1 we get˜ q = 1 + s + ( n − νn + 1 · n + ( n − s + nν )1 − s · q . We omit details of the calculation. This finishes the period and gives rise to an n -template.qP(q) ϕ = · · · = ϕ n +1 = νϕ = sϕ = ϕ = γϑ ϕ ϕ = · · · = ϕ n +1 P P P P = · · · = P n +1 P = · · · = P n +1 P − n − n − +1+1 +1 − nq ˜ q ˜ q ˜ q q Figure 4: Sketch period of P s,ν We again easily verify the claimed upper and lower limits ϕ j , ϕ j of the theorem. In-serting ϕ = s, ϕ n +1 = ν and for ϕ from (53), a calculation verifies equality in (SS2).Extending the interval for b ω as in (10) works similarly as in Section 5.7 by splitting theinterval [˜ q , ˜ q ] suitably to attain given ϕ within a corresponding range, we skip details.To estimate the Hausdorff and packing dimensions in Theorem 6.1, we evaluate thelocal contraction rates of P = P s,ν within the period interval [ q , q ] as δ ( P , q ) = n, if q ∈ [ q , ˜ q ] , , if q ∈ [˜ q , ˜ q ] , , if q ∈ [˜ q , q ] . By a similar argument as in Theorem 5.1 we see that to find the lower limit we mayconsider the average contraction rate within the interval [ q , q ] and finddim H (Σ ∗ ( n ) s,ν ) ≥ δ ( P ) = R q q δ ( P , q ) dqq − q = n (˜ q − q ) + (˜ q − ˜ q ) q − q = ˜ q + ( n − q − nq q − q = ˜ q q + ( n − ˜ q q − n q q − q q . Inserting for the ratios ˜ q /q , ˜ q /q , q /q from (57), (58), (59) the bound becomes D in(28) after tedious rearrangements, verifying (30).We finally estimate the packing dimension. Define ˜ q ,N , ˜ q ,N within [ q N , q N +1 ] corre-sponding to ˜ q , ˜ q in [ q , q ] likewise as in the proof of Theorem 5.1. Thendim P (Σ ∗ ( n ) s,ν ) ≥ δ ( P ) ≥ max { U, V } where U resp. V are the average limit contraction rates in the intervals [ q , ˜ q ,N ] resp.[ q , ˜ q ,N ] as N → ∞ . We show U ≥ E, V ≥ F with E, F from (29). Since we identified δ ( P ) as the average contraction rate in any interval [ q , q N ] and q = o ( q N ) as N → ∞ ,we evaluate U = lim N →∞ R ˜ q ,N q δ ( P , q ) dq ˜ q ,N − q = lim N →∞ δ ( P )( q N − q ) + n (˜ q ,N − q N )˜ q ,N − q = lim N →∞ δ ( P ) q N + n (˜ q ,N − q N )˜ q ,N = n + ( δ ( P ) − n ) · lim N →∞ q N ˜ q ,N and since q N / ˜ q ,N is independent of N , inserting δ ( P ) ≥ D this equals U ≥ n + q ˜ q ( δ ( P ) − n ) ≥ n + ( D − n ) q ˜ q = n + ( D − n ) ( n + 1)( s + ( n − ν + 1)(1 − s )( n + 1 + s − ν ) = E, where we used (57), (58), (59) to evaluate q / ˜ q . For V a similar calculation shows V = lim N →∞ R ˜ q ,N q δ ( P , q ) dq ˜ q ,N − q = lim N →∞ R q N +1 q δ ( P , q ) dq − R q N +1 ˜ q ,N δ ( P , q ) dq ˜ q ,N − q = lim N →∞ ( q N +1 − q ) δ ( P )˜ q ,N − q = δ ( P ) lim N →∞ q N +1 ˜ q ,N = δ ( P ) q ˜ q ≥ D q ˜ q = D n + 1 n + 1 + 2 s + ( n − ν = F, as claimed, where we used (58) in the last line. Theorem 6.1 is proved.6.3. Non-existence in Theorem 4.2.
To complete the proof of Theorem 4.2, the fol-lowing remains to be proved.
Theorem 6.2.
With the notation of Theorem 4.2, for ν / ∈ [ − s/n, ν ] the set Σ ( n ) s,ν isemtpy, i.e. there is no ξ inducing equality in (SS2) and with ϕ = s, ϕ n +1 = ν . For ν < − s/n again we get a contradiction to (11). So it remains to exclude ν > ν .This can be done very similarly as excluding µ < µ in Theorem 5.2 by some dual setup.Again using the method from the proof of [19, (7)] one can show that for arbitrarily large q , up to o ( q ) as q → ∞ , we must have a situation as in the interval [˜ q , q ] in Figure 4.Finally, for ν > ν we again derive a contradiction when considering the next meetingpoint of L , L to the left of ˜ q . We leave the details to the reader. Again we can deduceequality in the dimension formulas for s < , ν = ν since then the entire combined graphmust essentially be built up from consecutive periodical patterns as in Figure 4. PTIMALITY OF TWO INEQUALITIES 29 Final comments relating to work of Bugeaud, Laurent and Roy
According to the comments below [2, Theorem 3], sharpness of (BL1), (BL2) arerespectively equivalent to optimality of certain systems of inequalities.Identify ξ ∈ R n with its projective image in P n ( R ). Let 0 ≤ d ≤ n − ω d = ω d ( ξ ) the supremum of the real numbers u for which there existinfinitely many rational linear subvarieties L ⊆ P n ( R ) such that dim( L ) = d and d ( ξ, L ) ≤ H ( L ) − − u , where H ( L ) is the Weil height of any system of Pl¨ucker coordinates of L and d ( A, B ) denotes the distance of two projective sets
A, B ⊆ P n ( R ). Then ω correspondsto the classical exponent ω , and ω n − to our ω ∗ . Translating [17, Proposition 3.1] to ourformalism, any ω d can be written as an expression involving certain ϕ j ( q ) via(60) 11 + ω d = lim sup q →∞ n − d − P n +1 j = d +2 ϕ j ( q ) n + 1 , ≤ d ≤ n − , upon the convention ω d = ∞ if the right hand side becomes 0.Analyzing the proof of (BL2) in [2], identity is equivalent to identities(61) ω = ω + b ω − b ω and(62) ω d +1 = ( n − d ) ω d + 1 n − d − , ≤ d ≤ n − . For general ξ ∈ R n , the left hand sides are bounded below by the right hand sides in (61),(62) for every 0 ≤ d ≤ n −
2, so we have inequalities. A special case of a result by Roy [17,Theorem 2.3] implies that for any reasonable choice of ω = ω , there exists ξ ∈ R n thatsimultaneously satisfies all identities in (62) (for d = 0 as well). In fact, Marnat [11]evaluated Hausdorff and packing dimensions of the corresponding sets of ξ ∈ R n with theaid of Theorem 3.2. However, the condition (61) on b ω remains. Indeed, the restricitonsfrom Theorem 2.2 show that we cannot have this additional identity in certain cases. Wemention that if (61) and an extension of (62) also valid for d = 0 hold, then we couldconclude b ω = 1 /n . In particular, if ω > /n then our vectors ξ in Theorem 2.2 do nothave these properties as it can be checked that they induce b ω > /n . Similarly, lookingat the proof of (BL1) in [2] we check that equality happens if and only if(63) ω d − = dω d + 1 ω d + d − , ≤ d ≤ n − , and(64) ω n − = ( b ω ∗ − ω ∗ ω ∗ + b ω ∗ . Again in general there are just inequalities in (63) and (64), with the right hand sides notexceeding the left. Again (63) can be satisfied for ξ by Roy’s [17, Theorem 2.3], howeverwe are still left with a condition on b ω ∗ . Again, if we impose (64) and an extension of(63) valid for d = n − b ω ∗ = n , so for ω ∗ > n the examples inTheorem 2.1 do not satisfy these properties. References [1] A.S. Besicovitch. Sets of fractional dimensions (IV): On rational approximation to real numbers,
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