Unconventional height functions in simultaneous Diophantine approximation

Simultaneous Diophantine approximation is concerned with the approximation of a point \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbf x\in \mathbb R^d$$\end{document}x∈Rd by points \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbf r\in \mathbb Q^d$$\end{document}r∈Qd, with a view towards jointly minimizing the quantities \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Vert \mathbf x - \mathbf r\Vert $$\end{document}‖x-r‖ and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$H(\mathbf r)$$\end{document}H(r). Here \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$H(\mathbf r)$$\end{document}H(r) is the so-called “standard height” of the rational point \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbf r$$\end{document}r. In this paper the authors ask: What changes if we replace the standard height function by a different one? As it turns out, this change leads to dramatic differences from the classical theory and requires the development of new methods. We discuss three examples of nonstandard height functions, computing their exponents of irrationality as well as giving more precise results. A list of open questions is also given.


Main results
Throughout, d ≥ 1 is fixed, and · denotes the max norm on R d . Note that if d = 1, then H lcm = H max = H min = H prod = H 0 , where H 0 ( p/q) = q.
We begin by recalling Dirichlet's theorem: Theorem (Dirichlet's Approximation Theorem) For each x ∈ R d , and for any Q ∈ N, there exists p/q ∈ Q d with 1 ≤ q ≤ Q d such that Corollary (Dirichlet's Corollary) For every x ∈ R d \Q d , Equivalently 3 , x − r n < ψ 1+1/d • H lcm (r n ) for some sequence Q d r n → x. (1.1) In what follows, we consider analogues of Dirichlet's Corollary when H lcm is replaced by one of the three height functions H max , H min , and H prod . 3 There is a subtle distinction here: the existence of infinitely many rational points satisfying a given inequality, versus the existence of a sequence of rational points satisfying the given inequality and tending to the given point x. This distinction is important in what follows because otherwise the function H min would behave pathologically, since there exists a bounded region containing infinitely many points r ∈ Q d satisfying H min (r) = 1. But we do not want to say that such a sequence is a sequence of "approximations" of a given point x unless the sequence actually converges to the point x (which could only happen if x has an integer coordinate).

Exponents of irrationality
Before getting down to the details of our main theorems, we first consider "coarse" analogues of Dirichlet's Corollary. Specifically, we determine what the appropriate analogue of the exponent 1 + 1/d which appears in the formula (1.1) should be for our nonstandard height functions. More precisely: Definition Given a height function H : Q d → N and a point x ∈ R d \Q d , the exponent of irrationality of x is Equivalently, ω H (x) is the supremum of all α ≥ 0 such that x − r n < ψ α • H (r n ) for some sequence Q d r n → x.
The exponent of irrationality of the height function H is the number We observe that Dirichlet's Corollary implies that ω d (H lcm ) ≥ 1 + 1/d. In fact, the reverse inequality is true (and well-known): This means that 1 + 1/d is the "best exponent" that can be put into formula (1.1).
We are now ready to state the following theorem regarding exponents of irrationality: Theorem 1.1 (Exponents of irrationality of H max , H min , and H prod ) Remark The inequalities min ≤ prod 1/d ≤ max ≤ lcm ≤ prod automatically imply that Theorem 1.1 shows that when d ≥ 3, all inequalities are strict except the last.
(When d = 2, the third inequality is also not strict.) It is also interesting to note that lim d→∞ ω d (H max ) = 1 = lim d→∞ ω d (H lcm ), so the second inequality is asymptotically an equality.

More precise results
We now prepare to state our main theorems. These theorems will answer the question of what the appropriate analogue of the function ψ 1+1/d should be for our nonstandard height functions. More precisely: Definition Given a height function H : Q d → N, a function ψ : N → (0, ∞), and a point x ∈ R d , let Equivalently, C H,ψ (x) is the infimum of all C ≥ 0 such that x − r n < Cψ • H (r n ) for some sequenceQ d r n → x.
A function ψ will be called Dirichlet on R d with respect to the height function H if C H,ψ (x) < ∞ for all x ∈ R d \Q d , uniformly Dirichlet if sup R d \Q d C H,ψ < ∞, and optimally Dirichlet if ψ is Dirichlet and C H,ψ (x) > 0 for at least one x ∈ R d \Q d .
(This terminology originally appeared in [2].) We observe that Dirichlet's Corollary implies that the function ψ 1+1/d is uniformly Dirichlet on R d with respect to the height function H lcm , and in fact that In fact, the function ψ 1+1/d is optimally Dirichlet on R d with respect to the height function H lcm , due to the existence of so-called badly approximable vectors, i.e. vectors x ∈ R d \Q d for which C H lcm ,ψ 1+1/d (x) > 0. Roughly, the statement that ψ 1+1/d is optimally Dirichlet should be interpreted as meaning that in formula (1.1), the function ψ 1+1/d cannot be improved by more than a multiplicative constant. This interpretation was made rigorous in [2, Theorem 2.6 and Proposition 2.7].
Example The function ψ 2 (q) = q −2 is uniformly and optimally Dirichlet on R with respect to the height function H 0 . This fact may be equivalently expressed as follows: (i) (ψ 2 is uniformly Dirichlet) There exists C > 0 such that for all x ∈ R, there exist infinitely many p/q ∈ Q such that |x − p/q| ≤ Cq −2 . (ii) (Optimality) There exist x ∈ R and ε > 0 such that |x − p/q| ≥ εq −2 for all but finitely many p/q ∈ Q.
Remark We will sometimes deal with functions ψ which are not defined for all natural numbers, but only for sufficiently large numbers. In this case, the formula (1.5) may be interpreted as referring to an arbitrary extension of ψ to N; it is clear that the precise nature of the extension does not matter.  4 3. If not, can one give a criterion for determining whether or not a given function is Dirichlet?
It turns out that to answer these questions, we must consider two cases. The first case is when either H ∈ {H min , H prod } or d ≤ 2. In this case, the situation is similar to the situation for the height function H lcm : there is a uniformly and optimally Dirichlet function, and it comes from the class of power law functions (ψ α ) α≥0 . Precisely: In the second case, namely when H = H max and d ≥ 3, the situation is much different. Specifically, when d ≥ 3 the height function H max has the following unexpected property: It possesses no "reasonable" optimally Dirichlet function. To state this precisely, we need to define the class of functions that we consider to be reasonable. A Hardy L-function is a function which can be expressed using only the elementary arithmetic operations +, −, ×, ÷, exponents, logarithms, and real-valued constants, and which is well-defined on some interval of the form (t 0 , ∞). 5 For example, for any C, α ≥ 0 the function is a Hardy L-function. We have the following: Then no Hardy L-function is optimally Dirichlet on R d with respect to the height function H max .
Remark The class of Hardy L-functions includes almost all functions that one naturally encounters in dealing with "analysis at infinity", except for those with oscillatory behavior.
This answers question 1 above, so we would like next to answer question 3. Namely, given d ≥ 3 and a Hardy L-function ψ, how does one determine whether or not ψ is Dirichlet on R d with respect to H max ? Our final theorem (Theorem 1.4) will be a complete answer to this question. However, since it is complicated to state, we approach this theorem by degrees. As a first approximation we give the following corollary, which considers the case of a single error term added to the function ψ ω d (H max ) : Then ψ is (non-optimally) Dirichlet on R d with respect to H max if and only if In particular, letting C = 0, we see that the function ψ ω d (H max ) is not Dirichlet on R d with respect to H max .
This corollary now provides us with motivation to state our final theorem. Let ψ be the function defined by (1.6) when C = dγ d log 2 (γ d )/8. We know that ψ is not Dirichlet (on R d with respect to H max ), but that for any function of the form φ ε (q) = q ε/ log 2 log(q) , the product φ ε ψ is Dirichlet. This suggests that there is a function φ which grows more slowly than any φ ε such that the product φψ is still Dirichlet. What function can we multiply by? As it turns out, if  For a version of Theorem 1.4 which goes slightly beyond Hardy L-functions, allowing ψ to be a member of any Hardy field which contains the exponential and logarithm functions and is closed under composition, see Proposition 6.1.

Techniques
The main technique of this paper is to generalize the correspondence between the continued fraction expansion of an irrational number and its Diophantine properties into higher dimensions. This is done by introducing the notion of a data progression corresponding to an irrational vector x, which is a mathematical object that encodes information about the continued fraction expansions of all of the coordinates of x. The Diophantine properties of x can then be related to properties of the corresponding data progression. For more details see 2.2.
In the case of the height function H max , this correspondence translates the question of which functions are Dirichlet into a question about whether data progressions satisfying certain inequalities exist. We answer this question by converting it into a question about whether certain differential equations have nonnegative solutions, leading to the concept of a recursively integrable function. This concept is interesting in its own right and we study it in detail in Sect. 5. In particular we give a complete characterization of which Hardy L-functions are recursively integrable (Proposition 5.7), which leads to the characterization of which functions are Dirichlet described above in Theorem 1.4.

Summary of the paper
Section 2 contains preliminary results which are used in the proofs of our main theorems. In Sect. 3 we prove Theorem 1.2, as well as demonstrating formulas (1.3) and (1.4). Section 4 provides a motivation for the first formula of Theorem 1.1 without giving a rigorous proof. Section 5 is devoted to defining and analyzing the class of recursively integrable functions, a class which is used in the proof of Theorems 1.3 and 1.4. In Sect. 6 we prove Theorems 1.3 and 1.4, as well as demonstrating formula (1.2). Finally, a list of open questions is given in Sect. 7.

Lemmas concerning continued fractions
We begin our preliminaries with two lemmas concerning continued fractions. The first states that for x ∈ R, the convergents of the continued fraction expansion of x provide the best approximations to x as long as one is willing to accept a multiplicative error term. 6 Hence the Diophantine properties of x essentially depend only on the denominators of these convergents. The second states that given any sequence of numbers increasing fast enough, there is a number x such that the denominators of the convergents of the continued fraction expansion of x are equal up to an asymptotic to the elements of this sequence. Together, the two lemmas say that from a (sufficiently coarse) Diophantine point of view, the properties of a number can be encoded by an increasing sequence of integers.
Remark This section is mostly interesting if x is an irrational number. However, since the implied constants are supposed to be independent of x, the results are nontrivial even when x is rational. Lemma 2.1 Fix x ∈ R, and let ( p n /q n ) N 0 be the convergents of the continued fraction expansion of x (so that N = ∞ if and only if x / ∈ Q). Then for every p/q ∈ Q, there exists n ∈ N so that q q n and x − p q x − p n q n (cf. Convention 3).
Before we begin the proof, we recall [4, Theorem 1] that if (a n ) N 0 are the partial quotients of the continued fraction expansion of x, then p n = a n p n−1 + p n−2 (2.1) q n = a n q n−1 + q n−2 (2.2) for all n ≥ 1. Here we use the convention that p −1 = 1 and q −1 = 0. In particular, the sequence (q n ) N 0 is strictly increasing and satisfies q n a n q n−1 . We recall also [4,Theorems 9 and 13] that for all 0 ≤ n < N , Proof Consider the set S = {p /q ∈ Q : q ≤ q}, and let p /q ∈ S be chosen to minimize |x − p /q |. Then q ≤ q and |x − p /q | ≤ |x − p/q|, so we may without loss of generality assume that p/q = p /q . In this case, p/q is a best approximation of the first kind in the sense of [4, p. 24 for some 1 ≤ n ≤ N and 1 ≤ a ≤ a n . We consider two cases separately: • Case 1: a ≥ a n /2. In this case, 2q ≥ a n q n−1 + q n−2 = q n .
On the other hand, by [4,Theorem 17], p n /q n is a best approximation of the second kind, and thus also a best approximation of the first kind. Since q ≤ q n , this gives completing the proof in this case. • Case 2: 1 ≤ a < a n /2. In this case, since p/q lies on the same side of x as p n /q n (cf. [4,Theorem 4] and [4, Lemma on p.14]), we have = a n p n−1 + p n−2 a n q n−1 + q n−2 − ap n−1 + p n−2 aq n−1 + q n−2 = a n − a [a n q n−1 + q n−2 ][aq n−1 + q n−2 ] (cf. [4, Theorem 2]) ≥ a n /2 q 2 n 1 q n−1 q n x − p n−1 q n−1 .
Since q ≥ q n−1 , this completes the proof in this case.

Lemma 2.2
Let ( q n ) N 0 be a (finite or infinite) sequence satisfying q n+1 ≥ 2 q n and q 0 = 1. Then there exists x ∈ R so that if ( p n /q n ) N 0 are the convergents of the continued fraction expansion of x, then Proof The proof will proceed by recursively defining a sequence of integers (a n ) N 1 and then letting x be the unique number in (0, 1) whose partial quotients are given by (a n ) N 1 . Note that once this process is completed, for every 1 ≤ M ≤ N the value of q M can be computed from (2.2) using only the data points (a n ) M 1 together with the initial values q −1 = 0, q 0 = 1. Thus in our recursive step, once we have defined (a n ) M 1 , we may treat (q n ) M 1 as also defined.
Let q M be given by (2.2). Then i.e. (2.5) holds when n = M. This completes the recursive step.

Data progressions
Fix d ≥ 1. In the previous section, we learned how the Diophantine properties of an irrational number x are encoded in the sequence of denominators of the convergents of the continued fraction expansion of x. Continuing with this theme, given an irrational point x ∈ R d \Q d we would like to find a structure which encodes the Diophantine properties of x. It turns out that the appropriate structure for this encoding is given by the following definition: Equivalently, the sequence We say that is a d-dimensional data progression if the following hold: Remark In the sequel, the notation introduced in this definition will be used without comment.
k=1 is a one-dimensional data progression if and only if i k = 1 for all k, and the sequence (A k ) ∞ k=1 is increasing and tends to infinity. The canonical example is the sequence (q k ) ∞ k=1 of denominators of convergents of an irrational number x ∈ R\Q.

Lemma 2.4 Fix
: . Let = log exp and let = − log ψ exp. Suppose that and are uniformly continuous and coordinatewise increasing.
In particular where the supremum is taken over all d-dimensional data progressions .

Remark
The maps x → and → x implicitly described in parts (i) and (ii) of Lemma 2.4, respectively, are in fact independent of and ψ, as can be easily seen from the proof of Lemma 2.4. On an intuitive level these maps are "rough inverses" of each other, but we do not make this rigorous.

Remark 2.5 If
∈ {max, min, prod}, then ∈ {max, min, sum} is uniformly continuous and coordinatewise increasing. If ψ is a Hardy L-function whose decay is no faster than polynomial, then is uniformly continuous and increasing (Lemma A.4). Thus for the situations considered in this paper, the hypotheses of Lemma 2.4 will be immediately satisfied. . Then Proof of (ii). Let 0) is defined. Now fix n ≥ 0, and suppose that k(i, n) has been defined. Let k(i, n + 1) be the smallest value of k such that (2) if such a value exists; otherwise let N i = n. Then by Lemma 2.2, there exists x i ∈ R satisfying are the convergents of the continued fraction expansion of x i . By (II) of the definition of a data progression, we have N i = ∞ for at least one i and thus x : Here the understanding is that if n i = N i , then k i = ∞ and b (i) Using the fact that and are uniformly continuous and coordinatewise increasing, we deduce that On the other hand, for each i such that k i = ∞ we have

Proof of Theorem 1.2 and formulas (1.3), (1.4)
We begin by reformulating Theorem 1.2 using Theorem 1.1: Let Var( ) and Av( ) denote the variance and mean (average) of a d-tuple , respec-

Claim 3.2 We have
The proof is divided into two cases: either ∈ {min, prod}, or = max and d = 2.
Proof if ∈ {min, prod}. To begin with, we observe that (If = prod, then this equation is simply a reformulation of (3.2); if = min, it follows from the fact that min( k ) ≤ Av( k ).) Rearranging gives By (2.7), the above equation implies that Combining with (3.5) completes the proof of (3.3). Now suppose that k ∈ K , and observe that Av Combining with (3.5) gives (3.4).

Proof if
= max and d = 2. In this case, k , and so rearranging gives By (2.7), the above equation implies that . Now suppose that k ∈ K , and observe that b and thus k | 2 , this equation is equivalent to (3.4). To complete the proof of Proposition 3.1, observe that K is infinite by (II) of Definition 2.3. Thus, it follows from Claim 3.2 that Var( k ) → −∞. But this contradicts the fact that the variance of a data set is always nonnegative.
Proof of Optimality. Let x 1 , . . . , x d ∈ R be badly approximable numbers, and let x = (x 1 , . . . , x d ). We claim that C H ,ψ β d (x) > 0, demonstrating the optimality of ψ β d . Indeed, for each r ∈ Q d , , which implies the desired result.

Interlude: Motivation for the value of ω d (H max )
Before jumping into the proof of Theorems 1.3 and 1.4, in this section we try to motivate the formula (1.2). Our approach is as follows: The notion of a "data progression" is very broad, but it is natural to expect that "worst-case-scenario" data progressions will behave somewhat regularly. In fact, we will prove a rigorous version of this assertion in Sect. 6. But for now, let's just see what happens if we restrict our attention to data progressions which behave regularly.
Definition A data progression is periodic if the map k → i k is periodic of order d, and geometric if A k = γ k for some γ > 1. The number γ is called the mutliplier.
Remark If a data progression is periodic, then the map {1, . . . , d} k → i k must be a permutation.
Remark It is shown in Sect. 6 that to determine which functions ψ are Dirichlet on R d with respect to H max , it is sufficient to consider data progressions which are eventually periodic (Claim 6.6) and asymptotically geometric (Claim 6.5).
Proof Since is periodic, we have {b Since γ k → ∞, this completes the proof.
Fix α ≥ 0. From Lemma 2.4, we know that ψ α is Dirichlet on R d with respect to H max if and only if C max, α ( ) < ∞ for every d-dimensional data progression . Now comes the heuristic part: let's figure out what happens if we consider only periodic geometric data progressions, rather than all data progressions.

Proposition 4.2
The following are equivalent: In light of Lemma 4.1, it suffices to prove the following:  In the sequel, the following corollary will be useful: The unique maximum of the function with equality if and only if γ = γ d ; rearranging gives the desired result.

The class of recursively integrable functions
In this section we introduce a class of functions to be used in the proof of Theorem 1.4, the class of recursively integrable functions. We say that f is recursively integrable if for some t 1 ≥ t 0 the differential equation has a solution g : [t 1 , ∞) → [0, ∞). The class of recursively integrable functions will be denoted R. A solution g of (5.1) will be called a recursive antiderivative of f (regardless of its domain and range).
However, unlike the class of integrable functions, the class R is not closed under scalar multiplication, Indeed, we have: Conversely, suppose that C > 1/4, and by contradiction suppose that g : But since C > 1/4, there exists ε > 0 such that y 2 − y + C ≥ ε for all y ∈ R. Thus If f is a function such that the limit lim x→∞ x 2 f (x) exists and is not equal to 1/4, then Lemmas 5.
i.e. G is a recursive antiderivative of F if and only if g is a recursive antiderivative of f . (5.2); then g is a recursive antiderivative of f . To complete the proof we must show that g is nonnegative. But (5.1) together with the inequality f ≥ 0 show that In particular g is decreasing. Since g is bounded from below, it follows that lim x→∞ g(x) exists. Applying (5.3) again, we see that this limit must equal 0. Since g is decreasing, this implies that g(x) ≥ 0 for all x.
Remark An alternative proof of Lemma 5.4 may be given by applying Lemma 5.5 to the class of constant functions.
Applying Lemma 5.5 repeatedly to Lemma 5.4 yields the following: Corollary 5.6 For each N ≥ −1 and C ≥ 0, the function Remark There is a resemblance between Corollary 5.6 and the following well-known theorem: For each N ≥ −1 and α ≥ 0, the function We next show that Corollary 5.6 can be used to determine whether or not f ∈ R whenever f is a Hardy L-function. We have f ∈ R or f / ∈ R according to whether the former or the latter holds.
The second assertion is of course a direct consequence of Corollary 5.6 and Lemma 5.2.
Proof Let N be the order of f as defined in [3, §4], and consider the function On the other hand, it is readily seen that g is a Hardy L-function of order ≤ N . So by [3,Theorem 3], there exists ε > 0 such that either for all x sufficiently large, or g(x) ≥ ε for all x sufficiently large.
One more fact about transformations preserving recursive integrability will turn out to be useful:

, ∞) is recursively integrable if and only if the function
is recursively integrable.
Proof If g is a recursive antiderivative of f , then g λ (x) = λg(λx) is a recursive antiderivative of f λ . Since f = ( f λ ) 1/λ , the backwards direction follows from the forwards direction.
We next discuss the robustness of the concept of recursive integrability. As we have seen, it is not preserved under scalar multiplication. In particular, the sum of two recursively integrable functions is not necessarily recursively integrable. However, there are certain functions which can be safely added to a recursively integrable function without affecting its recursive integrability.
In what follows, H denotes a Hardy field (cf. Appendix 1) which contains the exponential and logarithm functions and is closed under composition. For example, H can be (and must contain) the class of Hardy L-functions described in the introduction.

Definition 5.9 A nonnegative function
Note that the sum of any two ignorable functions is ignorable. Moreover, if f 2 is ignorable and 0 ≤ f 1 ≤ f 2 , then f 1 is ignorable (assuming f 1 ∈ H). By Archimedes' principle, it follows that the class of ignorable functions is closed under (nonnegative) scalar multiplication.
Proof Fix f 1 ∈ R ∩ H, and let g 1 : [t 1 , ∞) → [0, ∞) be a recursive antiderivative of f 1 . Fix C > 1/ε, and let Since f 1 ∈ H, we have either Then is a recursive antiderivative of f 3 . But by (5.7), we have f 3 ≥ 0. By the argument used at the end of the proof of Lemma 5.5, the limit lim x→∞ g 3 (x) exists and is equal to zero. Equivalently, this means that xg 1 (x) → 1/2 as x → ∞. Thus Since C > 1/ε, this implies that for all sufficiently largex.
Remark Applying Lemma 5.5 repeatedly shows that for all N ≥ −1 and ε > 0 the function We finish this section by providing a number of equivalent conditions to the recursive integrability of a function f ∈ H. The following proposition should be thought of as an analogue of the Integral Test which says that a increasing function f : [0, ∞) → [0, ∞) is integrable if and only if the series ∞ k=1 f (k) is summable. It should be noted that as with the Integral Test, the motivation here is not to determine whether a function is recursively integrable by using an equivalent condition, but rather to determine whether one of the equivalent conditions is true by determining whether the function in question is recursively integrable.

Proposition 5.11
Suppose f ∈ H is nonnegative. Then for any t ∈ R, the following are equivalent: There exists a nonnegative sequence (S k ) k≥k 0 satisfying S k → 0 and Remark Suppose that f satisfies any of the conditions (B1)-(C2). Plugging the formula S k → 0 into the appropriate Eqs. (5.8) or (5.10) shows that lim sup k→∞ f (k) ≤ 0. Since f ∈ H and f ≥ 0, it follows that f (x) → 0 as f → ∞. Again using the facts that f ∈ H and f ≥ 0, we deduce that f is decreasing for sufficiently large x. Similar reasoning applies if we assume that f satisfies (A). Thus in the proof of Proposition 5.11, we may assume that f is decreasing on its domain of definition. = 0. Backwards induction shows that for each k, the sequence (S (N ) k ) N ≥k is increasing, and S Then the nonnegative sequence ( S k ) k≥k 0 satisfies (5.9).
Proof By (5.9) and (5.12), we have In analogy with the proof of Lemma 5.10, for each k let T k ≥ −1/2 satisfy Plugging into (5.13) gives It follows that the sequence (T k ) ∞ 1 is decreasing and bounded from below. Thus the limit lim k→∞ T k exists, and which implies that lim k→∞ T k = 0. Equivalently, lim k→∞ kS k = 1/2.
In particular, S k → 0. Fix C > 0, and let Then S k → 0 as well. So to complete the proof, we need to show that (5.10) holds for the sequence ( S k ) k≥k 0 . We have So to complete the proof, it suffices to show that if C is large enough, then E k ≥ 0 for all k sufficiently large. And indeed, Thus by choosing C > t/8 + 5/4, we complete the proof.
Proof of (C1) ⇒ (C2). Suppose that the sequence (S k ) k≥k 0 satisfies S k → 0 and (5.10). Fix k 1 ≥ k 0 large enough so that Let ( S k ) k≥k 1 be the unique sequence satisfying (5.11) and S k 1 = S k 1 . An induction argument shows that for all k ≥ k 1 , S k ≤ S k ≤ S k 1 and S k+1 ≤ S k . In particular the sequence ( S k ) k≥k 0 is nonnegative. To complete the proof we need to show that S k → 0. Since ( S k ) k is decreasing, the limit L = lim k→∞ S k exists. Taking the limit of (5.11) we find that Since 0 ≤ L ≤ S k 1 , this implies that L = 0.
so by Remark 5.3 f ∈ R. Now suppose t = 0 and that (S k ) k≥k 0 satisfies S k → 0 and (5.10). Let be large enough so that and let k 1 ≥ k 0 be large enough so that S k 1 + ≤ 2/ . Then an induction argument shows that In particular, there exists C > 0 such that S k ≤ C/k for all k ≥ k 0 . Then and so by the t = 0 case of (C1) ⇒ (A), the function x → f (x) − |t|C 3 /x 3 is recursively integrable. Since the function x → |t|C 3 /x 3 is ignorable (Lemma 5.10), f is also recursively integrable.

Proof of Theorems 1.3 and 1.4 and formula (1.2)
As in Sect. 5, H denotes a Hardy field which contains the exponential and logarithm functions and is closed under composition, for example the field of Hardy L-functions. As in Sect. 4, we write   The proof of Proposition 6.1 will be divided into three parts: the proof of (D) ⇒ (B), which constitutes the hardest part of the argument; the proof of (C) ⇒ (D), which is essentially the proof of (D) ⇒ (B) in reverse, but made easier due to the explicitness of the data structure in question; and finally, the reduction of the theorem to those two implications, which is essentially a corollary of Lemma 5.10.
Remark Throughout the proof we will assume that x 2 for all x sufficiently large. (6.1) The justification of this assumption follows along the same lines as Remark 5.12. Specifically, suppose that Proposition 6.1 holds whenever ψ satisfies (6.1). Let ψ − and ψ + denote the functions for which equality holds in the left and right hand inequalities of (6.1), respectively. Then by Lemma 5.4, ψ + satisfies (A)-(D) of Proposition 6.1 while ψ − fails to satisfy them. Now let ψ ∈ H be arbitrary. If ψ does not satisfy (6.1), then by Lemma A.2 either ψ(q) ≥ ψ + (q) for all q sufficiently large or ψ(q) ≤ ψ − (q) for all q sufficiently large. In the first case, we have C H max ,ψ ≤ C H max ,ψ + and so (A)-(D) of Proposition 6.1 hold. In the second case, we have C H max ,ψ ≥ C H max ,ψ − and so (A)-(D) of Proposition 6.1 fail to hold.
Remark 6.2 When reading the proof of (D) ⇒ (B), one should check that the implications (6.3) ⇒ (6.4) ⇒ (6.5) ⇒ (6.10) are all invertible if one assumes the following facts about : max( k ) = A k for all k ∈ N, and is eventually periodic in the sense of Claim 6.6. The converse directions will be used in the proof of (C) ⇒ (D).

Notation
The following notations will be used in the course of the proof: Note that according to these notations,

Proof of (D) ⇒ (B)
We prove the contrapositive. Suppose that sup R d \Q d C H max ,ψ = ∞, and we will show that f ψ ∈ R. By Lemma 2.4, we have sup C max, ( ) = ∞, where the supremum is taken over d-dimensional data progressions . In particular, there exists a d-dimensional data progression for all k sufficiently large.

Claim 6.3
We may suppose without loss of generality that max( k ) = A k for all k ∈ N.
Proof Consider the set K = {k ∈ N : max( k+1 ) > max( k )}. The set K is infinite by part (II) of the definition of a data progression. Let (k ) ∞ 1 be the unique increasing indexing of K , and consider the data progression = (max( k ), i k ) ∞ =1 . Note that for all ∈ N and i = 1, . . . , d, .
Plugging all these into (6.3) gives i.e. (6.3) holds for the data progression .
So in what follows, we assume that max( k ) = A k for all k ∈ N. Using this fact together with (2.6), (6.3) becomes Letting t k = A k+1 /A k , we may rewrite the above equation as using (2.6), (6.4) then becomes for all k sufficiently large.
Proof We clearly have f k ≤ 1 for all k, so by Claim 6.4, the sequence ( f k ) ∞ 1 converges to a positive number. Thus f k f k+1 → 1. Combining with (6.6), we see that (6.7) Proof Combining (6.4) and Claim 6.5, we see that On the other hand, for each j = 0, . . . , d − 2, by Claim 6.5 we have Dividing both sides by (A k ) d finishes the proof.

Corollary 6.8 For all k,
Proof By Claim 6.5, Taking d 2 th roots completes the proof.
Using Corollary (6.7), (6.5) becomes Writing s k = t k /γ d − 1, a few arithmetic calculations show that the above inequality is equivalent to Consequently, it becomes important to study behavior the function near the origin. We calculate the gradient and Hessian of f at 0: Since f (0) = 0, this means that f can be estimated in a neighborhood of the origin by the formula In fact, we can be explicit: (6.11) holds whenever x ≤ 1/2. Continuing with the proof, for k ∈ N let (cf. (6.2)).

Claim 6.9
For all k sufficiently large, Proof Since f ψ ∈ H, we may differentiate the inequalities (6.1) (cf. Lemma A.3) to get x 3 for all x sufficiently large. (6.12) Using (6.2) and applying the fundamental theorem of calculus, we have 1 k 3 · (by Claim 6.5 and Corollary 6.8) Now fix C 1 > 0 large to be determined, then fix δ > 0 small to be determined (possibly depending on C 1 ), and finally fix k 0 ∈ N large to be determined (possibly depending on both δ and C 1 ). Let (S k ) ∞ k=k 0 be the unique sequence defined by the equations The following claim is the heart of the proof: Claim 6.10 If k 0 and C 1 are sufficiently large and δ is sufficiently small (with k 0 allowed to depend on δ, which is in turn allowed to depend on C 1 ), then for all k ≥ k 0 .
Proof Throughout the proof, we will assume that δ < 1/ max(2, C 1 ) and that k 0 ≥ 4C 1 . Since δ and k 0 are both allowed to depend on C 1 , these assumptions are justified.
In particular, the rightmost inequality of (6.14) requires no proof. By Claim 6.5, we have s k → 0. Thus, the leftmost inequality of (6.14) can be achieved simply by an appropriate choice of k 0 .
The proof of the two middle inequalities of (6.14) is by strong induction on k.
For this part of the proof, we'll think of C 1 , δ > 0 as being fixed. Define the sequence (T j ) d−2 j=0 via the formula Since δ < 1/ max(2, C 1 ), the sequence (T j ) d−2 j=0 is strictly decreasing and strictly positive. Note that for each j = 0, . . . , d − 2, where the superscript of k 0 is merely making explicit the fact that the sequence (S k ) k≥k 0 depends on k 0 . On the other hand, So if k 0 is sufficiently large, then (6.14) holds for k = k 0 + j.
In particular, plugging in j = 1 and using the induction hypothesis, we see that the third inequality of (6.14) holds for k = . So to complete the proof, we need only to demonstrate that the second inequality of (6.14) holds for k = .
Remark We emphasize that here and below, the implied constants of asymptotics may not depend on C 1 , δ, or k 0 .
Proof Since ≥ k 0 ≥ C 1 , we have 1/ ≤ 1/C 1 . On the other hand, by Subclaim 6.12 and the induction hypothesis we have |S | max 1 2 , |S −1 | ≤ 1 C 1 · Definition 6.14 For the purposes of this proof, an expression will be called negligible if its absolute value is less than a constant times a 3 . (The constant must be independent of C 1 , δ, and k 0 .) We'll write A ∼ B if the difference between two expressions A and B is negligible.
We are now ready to continue our calculation: φ (by (6.10)) φ (by (6.11)) Let C 5 > 0 be the implied constant. Combining with (6.13) shows that for all sufficiently large k. By Proposition 5.11, the function is recursively integrable. By Lemma 5.10, it follows that f ψ ∈ R.

Proof of (C) ⇒ (D)
As before, we will prove the contrapositive. Suppose that f ψ ∈ R, and we will show that sup R d \Q d C H max ,ψ > 0. Fix C 1 > 0 large to be determined. By Lemma 5.10, the function x → f ψ (x) + C 1 /x 3 is recursively integrable. Thus by Proposition 5.11, there exists a nonnegative sequence (S k ) k≥k 0 satisfying Let i k = k (mod d), and consider the d-dimensional data progression = (A k , i k ) ∞ k=k 0 . Since the sequence (A k ) ∞ k 0 is increasing, Remark 6.2 applies and we have the implication (6.10) ⇒ (6.3). Note that if (6.3) holds for all k sufficiently large, then we are done, as C max, ( ) ≥ 0 and then Lemma 2.4 completes the proof.
Let us proceed to demonstrate (6.10). We begin by reproving Subclaims 6.11, 6.12, and 6.13 in our new context. Fix k ∈ N. The inequality S k+1 ≤ S k is immediate from (6.17). If k is sufficiently large, then f ψ (k) ≤ 1/k 2 , k ≥ C 1 , and S k ≤ 1/C 1 , so This implies that S k max(1/k 2 , S k+1 ), completing the proof of the analogue of Subclaim 6.12. Finally, let a k = max(1/k, S k ); it is immediate that a k ≤ 1/C 1 if k is sufficiently large.
As in the proof of Claim 6.10 we call an expression A negligible if |A| a 3 k , and write A ∼ B if A − B is negligible. The argument following Definition 6.14 shows questions can be asked if "every" is replaced by "almost every"-with respect to Lebesgue measure or even with respect to some fractal measure. Once we know what "almost every" point does, it can be asked what is the Hausdorff dimension of the set of exceptions, i.e. the set of x which behave differently from almost every point. In the case of the height function H lcm , such questions have been extensively studied. Thus, the next step in producing a Diophantine theory of the height functions H max , H min , and H prod similar to that for H lcm would be to answer the following questions: Question 7.1 (Analogue of Khinchin's theorem) Fix ∈ {max, min, prod}, and let ψ be a Hardy L-function. Must the sets {x ∈ R d : C H ,ψ (x) = 0} and {x ∈ R d : C H ,ψ (x) < ∞} be either null sets or full measure sets? If so, which one? Can the same theorem be proven with a weaker assumption than ψ being a Hardy L-function (for example, assuming only that ψ is decreasing)?