On quantitative aspects of the unit sum number problem

Clemens Fuchs, Robert Tichy and Volker Ziegler

Abstract. We investigate the function u_{K,S}(n; q), which counts the number of representations of algebraic integers α with |N_{K/ℚ}(α)| ≤ q for some real positive q, so that they can be written as sum of exactly n S-units of the number field K.

Mathematics Subject Classification (2000). 11N45,11D45. Keywords. sums of units, unit sum number problem.

1. Introduction

The study of the additive structure of units in rings originates from investigations due to Zelinsky [15] back in the fifties of the previous century. Zelinsky showed that every endomorphism of a vector space V over a division ring D can be written as the sum of two automorphisms, unless D is the field with two elements and V is of dimension 1. Similar results were obtained for other endomorphism rings (for an overview see [14]).

The additive unit structure of maximal orders of number fields has been studied, too. In particular, Jacobson [10] observed that every element in the maximal orders of the number fields ℚ(\sqrt{2}) and ℚ(\sqrt{5}) can be written as the sum of distinct units. Later all quadratic, cubic and quartic fields with this property were determined (see [12, 2, 3]). By a combination of results on S-unit equations and combinatorial results Jarden and Narkiewicz [11] showed that given a finitely generated domain B of characteristic zero then for every positive integer k there exists an α ∈ B such that α can not be written as the sum of k units. In view of this observation two problems arise:

The qualitative problem: Which ring of integers are generated by their units?

The third author is supported by the Austrian Science Fund FWF (project numbers: S9611 and S9603).

The quantitative problem: How many non-associated algebraic integers with

bounded norm, height, etc. can be written as the sum of k units? The qualitative problem was solved for quadratic and complex cubic fields (see [2, 1, 13]). Also some special classes of quartic fields were investigated (see [9, 16]). Recently the quantitative problem was investigated for quadratic fields and some purely quartic fields (see [9, 8]).

This paper is devoted to the quantitative problem. Let K be a number field, N_{K/ℚ} the field norm and S a finite set of places of K including the archimedian ones. We denote by O_{K,S} = {α ∈ K : |α|_{ν} ≤ 1 for all ν ∉ S} the ring of S-integers and its unit group by U_{K,S}; the group of S-units. As usual two S-integers α and β of K are said to be associated if there exists an S-unit ϵ such that α = βϵ and we write α ∼ β. In order to give precise statements we introduce the following counting function:

Definition 1. The counting function u_{K,S}(n; q) = u(n; q) is defined as number of equivalence classes [α]_{∼} such that

N(α) := ∏_{ν∈S} |α|_{ν} ≤ q,  α = ∑_{i=1}^{n} ϵ_{i},  ϵ_{i} ∈ U_{K,S}

and no subsum of ϵ_{1} + ϵ_{2} + … + ϵ_{n} vanishes.

Note that the function u(n; q) is well defined, since N(α) = N(β) if α ∼ β. Before we state our main theorem we fix the following notation for the rest of the paper. Let |S| = s+1. Then the group U_{K,S} of S-units of K is a free abelian group with s generators (see [4]) and we fix a fundamental system ϵ_{1}, …, ϵ_{s} of S-units generating it. By ω_{K} and Reg_{K,S} we will denote the number of roots of unity and the S-regulator of K, respectively.

Theorem 1. Let ϵ > 0. Under the above assumptions we have

u(n; q) = \frac{c_{n−1,s}}{n!} (\frac{ω_{K}(log q)^{s}}{Reg_{K,S}} )^{n−1} + o((log q)^{(n−1)s−1+ϵ})

as q → ∞, where the constant c_{n,s} is the volume of

{(x_{11}, …, x_{ns}) ∈ ℝ^{ns} : h(x_{11}, …, x_{ns}) < 1}

with

h(x_{11}, …, x_{ns}) = ∑_{i=1}^{s} max{0, x_{1i}, …, x_{ni}} + max {0, − ∑_{i=1}^{s} x_{1i}, …, − ∑_{i=1}^{s} x_{ni}} .

Our theorem is closely related to results due to Everest [5, 6] who counted the number of solutions to certain norm inequalities. However, the full result of Everest makes heavy use of many non-elementary results such as theorems on the uniform distribution of sequences, Schmidt’s subspace theorem, Baker’s theory on linear forms in logarithms, etc. Using the full result would lead us to an error term without ϵ and a big-O instead of a small-o, where we use O and o with their usual meaning.

In order to give the reader more insights we sketch Everest’s proof and try to avoid most of the non elementary steps. Unfortunately, we cannot avoid an application of Schmidt’s subspace theorem (see Lemma 3, second statement). However, the interested reader is advised to study Everest’s papers for a full account to the techniques described below.

In the next section we introduce some notations that will help us (in Section 3) to state Everest’s results and to sketch his proofs. Applying these results we will prove our main theorem in Section 4. Since the definition of the constant c_{n,s} is not very illuminating we discuss a formula for c_{n,s} in the last section. This also enables us to compare the result with previous results of this form (particularly with the results in [8]).

2. Some Notations

We want to study S-unit representations with small norm and so we introduce the following functions. We will use the definition

∗U_{K,S}^{n} = \{\underline{x} = (1, x_{2}, …, x_{n}) ∈ U_{K,S}^{n} : no subsum of 1 + x_{2} + ⋯ + x_{n} vanishes\}.

Let \underline{x} = (1, x_{2}, …, x_{n}) ∈ U_{K,S}^{n}. Then consider

N_{K,S}(\underline{x}) = N(\underline{x}) = ∏_{ν∈S} |1 + x_{2} + ⋯ + x_{n}|_{ν}

and its related counting function

N_{K,S}(q) = N(q) = ♯ \{\underline{x} ∈ ^{∗}U_{K,S}^{n} : N(\underline{x}) < q\} .

We are also interested in the following variant: Let \underline{c} = (c_{1}, …, c_{n}) ∈ (K^{∗})^{n} be fixed, then we define

N\underline{c}(\underline{x}) = ∏_{ν∈S} |c_{1} + c_{2}x_{2} + ⋯ + c_{n}x_{n}|_{ν}

and the corresponding counting function N\underline{c}(q) by the number of \underline{x} ∈ U_{K,S}^{n} such that N\underline{c}(\underline{x}) < q and no subsum of c_{1} + c_{2}x_{2} + ⋯ + c_{n}x_{n} vanishes.

As we will see in the next section these functions are closely related to the height functions

H_{ν}(\underline{x}) = max\{|x_{i}|_{ν} : i = 1, …, n\}

and

H(\underline{x}) = ∏_{ν∈S} max\{|x_{i}|_{ν} : i = 1, …, n\} = ∏_{ν∈S} H_{ν}(\underline{x})

and their counting function

H(q) = ♯ \{\underline{x} ∈ U_{K,S}^{n} : H(\underline{x}) < q\} .

As will be shown later the following subset U_{0} ⊂ U_{K,S}^{n} will yield the main contribution to H(q): We write H_{ν}^{∗} to be the second largest member of the set {|x_{i}|_{ν} : i = 0, …, n}. Given real numbers A_{ν} > 0, B_{ν} > 1 we define

U_{0} = \{\underline{x} ∈ ^{∗}U_{K,S}^{n} : ∀ν ∈ S, \frac{H_{ν}^{∗}(\underline{x})}{H_{ν}(\underline{x})} < A_{ν}e^{−B_{ν} log log H(\underline{x})}\} .

Next we define the counting function

H_{0}(q) = ♯ \{\underline{x} ∈ U_{0} : H(\underline{x}) < q\} .

Assume for the rest of the paper that A_{ν} and B_{ν} are fixed numbers. The particular choice of A_{ν} and B_{ν} does not effect the asymptotics but rather the constants in the error terms, which are not the subject of interest in our considerations.

3. Sums of S-units

In this section we adapt the results and techniques due to Everest [5, 6] to our case. We start with

Lemma 1. For q → ∞ we have

H(q) = c_{n−1,s} (\frac{ω_{K}(log q)^{s}}{Reg_{K,S}} )^{n−1} + O((log q)^{(n−1)s−1}).

Proof. Let S = {ν_{1}, …, ν_{s+1}} and x_{1} = 1, x_{2}, …, x_{n} be S-units. Write x_{i} = ζ_{i}ϵ_{1}^{k_{i1}} ⋯ ϵ_{s}^{k_{is}} for i = 2, …, n. Taking logarithms of the absolute values | · |_{ν} with ν ∈ S and recalling the definition of H(\underline{x}) we see that the function H(q) essentially counts the number of points \underline{k} = (k_{21}, …, k_{ns}) ∈ ℤ^{(n−1)s} that fulfill the inequalities

∑_{j=1}^{s+1} max\{L_{i}^{(j)}(\underline{k}) : i = 1, …, n\} < log q  (1)

for i = 1, …, n, where L_{1}^{(j)}(\underline{k}) = 0 and

L_{i}^{(j)}(\underline{k}) = k_{i1} log |ϵ_{1}|_{ν_{j}} + ⋯ + k_{is} log |ϵ_{s}|_{ν_{j}} = log |x_{i}|_{ν_{j}}

for i = 2, …, n, and for j = 1, …, s + 1. A linear transformation of determinant Reg_{K,S}^{−1} turns the L_{i}^{(j)}(\underline{k}) into

L̃_{i}^{(j)}(\underline{k}) = \{ \begin{matrix}k_{ij} & for j ≤ s \\ − ∑_{l=1}^{s} k_{il} & for j = s + 1. \end{matrix}

Note that the number of lattice points in a polyhedron of dimension n is its volume plus an error term of the size about the n − 1 dimensional volume of its faces. Bearing in mind the linear transformation and the fact that the number of lattice points are points counted by H(q) up to multiplication by roots of unity we obtain the lemma. □

For the function H_{0}(q) we obtain (see [6, Proposition 1]):

Lemma 2. For all ϵ > 0 and q → ∞ we have

H_{0}(q) = H(q) + o ((log q)^{(n−1)s−1+ϵ}) .

Note that the proof of this lemma does not need any deep results and can be achieved by considering Dirichlet series (see also [5, Lemma 3]).

Next, we want to establish a relation between H(\underline{x}) and N(\underline{x}) (see [5, Lemma 6]).

Lemma 3. 1. For all \underline{x} ∈ U_{0} we have log N(\underline{x}) = log H(\underline{x}) + O (1/ log(H(\underline{x}))).

2. For all \underline{x} ∈ ^{∗}U_{K,S} there exists a constant θ > 0 such that log N(\underline{x}) > θ log H(\underline{x}).

Proof. We have

log N(\underline{x}) = ∑_{ν∈S} log |1 + x_{2} + ⋯ + x_{n}|_{ν}

= ∑_{ν∈S} log H_{ν}(\underline{x}) + ∑_{ν∈S} log (1 + O (\frac{H_{ν}^{∗}(\underline{x})}{H_{ν}(\underline{x})})) .

Because of the definition of U_{0} the first statement is proved. The second statement is a consequence of Schmidt’s subspace theorem (see [7, Theorem 2] and in the case of K a quadratic field see also [8, Lemma 3]) □

Now we establish the main result of this section (see also [6, Theorem 1]):

Proposition 1. For all ϵ > 0 and q → ∞ we have

N(q) = c_{n−1,s} (\frac{ω_{K}(log q)^{s}}{Reg_{K,S}} )^{n−1} + o((log q)^{(n−1)s−1+ϵ}).

Proof. We break up the counting function into two parts:

N(q) = ♯\{\underline{x} ∈ ^{∗}U_{K,S} : N(\underline{x}) < q, x ∈ U_{0}\} + ♯\{\underline{x} ∈ ^{∗}U_{K,S} : N(\underline{x}) < q, x ∉ U_{0}\}
= N_{1}(q) + N_{2}(q)

For the second part we obtain by Lemma 3

N_{2}(q) = ♯\{\underline{x} ∈ ^{∗}U_{K,S} : H(\underline{x})^{θ} < q, x ∉ U_{0}\} = o ((log q)^{(n−1)s−1+ϵ}) ,

where the last equality is obtained by a combination of the Lemmas 1 and 2. Since those \underline{x} that satisfy H(\underline{x}) < (log q)^{1/2} are only O((log q)^{(n−1)s/2}) in number, we may assume H(\underline{x}) > (log q)^{1/2}. So we obtain

N_{1}(q) = ♯\{\underline{x} ∈ U_{0} : log(H(\underline{x})) < log q + O(1/\sqrt{log q})\}
= H_{0} (q^{1+O(1/\sqrt{log q})})
= H (q^{1+O(1/\sqrt{log q})}) + o (((1 + O(1/\sqrt{log q})) log q)^{s(n−1)−1+ϵ})
= c_{n−1,s} (\frac{ω_{K}(log q)^{s}}{Reg_{K,S}} )^{n−1} + o((log q)^{(n−1)s−1+ϵ}) + O(log q^{(n−1)s−3/2})

= c_{n−1,s} (\frac{ω_{K}(log q)^{s}}{Reg_{K,S}} )^{n−1} + o((log q)^{(n−1)s−1+ϵ}).

Thus the statement follows. □

Corollary 1. Assume for all ν ∈ S that log |ϵ_{1}|_{ν}, …, log |ϵ_{s}|_{ν} generate a ℚ-space of dimension at least 2. Then

N(q) = c_{n−1,s} (\frac{ω_{K}(log q)^{s}}{Reg_{K,S}} )^{n−1} + O((log q)^{(n−1)s−1}).

Proof. Combine Everest’s result [6, Theorem 1] with Proposition 1. □

At the end of the section we want to state Everest’s result [6, Theorem 1] for N\underline{c}(q) which stated in this form may be obtained analogously to the proof of Proposition 1, i.e. the only non elementary part is the proof of the second statement of Lemma 3.

Proposition 2. For all \underline{c} ∈ (K^{∗})^{n} we have

N\underline{c}(q) = O((log q)^{(n−1)s}).

Now we have collected all results that are needed to prove the main theorem (see next section).

4. The number of S-unit representations

This section is devoted to the proof of our main theorem (Theorem 1). Assume α = x_{1} + … + x_{n} can be written as a sum of S-units, where

x_{i} = ζ_{i}ϵ_{1}^{k_{i1}} ⋯ ϵ_{s}^{k_{is}}

for i = 1, …, n. Then we can write α = 1 + x_{2} + ⋯ + x_{n} with k_{11} = … = k_{1s} = 0, k_{i1} ≤ ⋯ ≤ k_{is} for i = 2, …, n and 0 = k_{1s} ≤ k_{2s} ≤ ⋯ ≤ k_{ns}, since we are interested only in equivalence classes of associated integers. Let us denote by u_{0}(n; q) the number of representations, where all units are pairwise distinct. Then we have u_{0}(n; q) = N(q)/n! − r, where r is the number of integers α that have two distinct representations as sums of n distinct S-units. As shown in [8, Lemma 2] r depends only on n and S but not on q. The factor 1/n! comes from the fact that the order of the x_{i} does not matter in the computation of u_{0}(n; q).

On the other hand we have

u(n; q) = u_{0}(n; q) + ∑_{\underline{c}} \frac{N\underline{c}(q)}{l(\underline{c})!} + O(1),  (2)

where the O(1) comes from the α’s with two distinct representations and the sum runs over all \underline{c} = (c_{1}, …, c_{m}) ∈ ℤ^{m} with c_{i} ≥ 1 and c_{1} + ⋯ + c_{m} = n and (c_{1}, …, c_{m}) ≠ (1, …, 1), i.e. m < n for all \underline{c}’s appearing in the summation. Moreover we write l(\underline{c}) = m. Applying the Propositions 1 and 2 to formula (2) proves our main Theorem 1.

5. Some results for c_{n,s}

In this section we want to discuss the quantity c_{n,s}. For small values of s and n it is possible to compute the quantity c_{n,s} directly. We did this for all pairs (n, s) with n + s ≤ 6 (see Table 1).

Table 1. Values for c_{n,s}

s⧹n  1  2  3  4  5
1  2  3  4  5  6
2  3  15/4  7/2 45/16
3  10/3  7/3 55/54
4  35/12 275/32
5  21/10

It is also possible to calculate all values of c_{n,s} for either n = 1 or s = 1. In general, we were not able (yet) to succeed in doing the necessary calculations. We can prove the following:

Lemma 4. We have

c_{n,1} = n + 1, c_{1,s} = \frac{1}{s!}\binom{2s}{s}

and

\frac{2^{ns}}{(ns)!} < c_{n,s} < 2^{ns}.

Proof. First, we consider the case s = 1, i.e. we have to compute the volume of

B_{n,1} := {(x_{1}, …, x_{n}) ∈ ℝ^{n} : max{0, x_{1}, …, x_{n}} + max{0, −x_{1}, …, −x_{n}} < 1} .

Assume that the following holds:

x_{1} > x_{2} > ⋯ > x_{r} > 0 > x_{r+1} > ⋯ > x_{n}

with 0 < r < n. Then we have x_{1} − x_{n} < 1. Note that the case r = 0, respectively r = n, yields the volume of a simplex with rectangular edges of length 1 and hence volume 1/n!. In the other cases the volume is

∫_{x_{1}=0}^{1} ∫_{x_{n}=x_{1}−1}^{0} ∫_{x_{2}=0}^{x_{1}} ⋯ ∫_{x_{r}=0}^{x_{r−1}} ∫_{x_{n−1}=x_{n}}^{0}⋯ ∫_{x_{r+1}=x_{r+2}}^{0}1 dx_{r+1} ⋯ dx_{n−1}dx_{r} ⋯ dx_{2}dx_{n}dx_{1}

= ∫_{x_{1}=0}^{1} ∫_{x_{n}=x_{1}−1}^{0} ∫_{x_{2}=0}^{x_{1}} ⋯ ∫_{x_{r}=0}^{x_{r−1}} \frac{(−x_{n})^{n−r−1}}{(n − r − 1)!} dx_{r} ⋯ dx_{2}dx_{n}dx_{1}

= ∫_{x_{1}=0}^{1} ∫_{x_{n}=x_{1}−1}^{0} \frac{(−x_{n})^{n−r−1}x_{1}^{r−1}}{(n − r − 1)!(r − 1)!} dx_{n}dx_{1}

= ∫_{x_{1}=0}^{1} \frac{(1 − x_{1})^{n−r}x_{1}^{r−1}}{(n − r)!(r − 1)!} dx_{1} = \frac{1}{n!}.

So any case (also the cases r = 0 and r = n) yields the volume 1/n!. Since there are (n + 1)! cases we obtain c_{n,1} = (n + 1)!/n! = n + 1.

Now we consider the case n = 1. In this case we have to compute the volume of

B_{1,s} := {(x_{1}, …, x_{s}) ∈ ℝ^{s} : h(x_{1}, …, x_{s}) < 1},

where h(x_{1}, …, x_{s}) := max{0, x_{1}}+⋯+max{0, x_{s}}+max{0, −x_{1} −⋯−x_{s}}. Let us assume x_{1}, …, x_{r} > 0 and x_{r+1}, …, x_{s} ≤ 0. Then h(x_{1}, …, x_{s}) = x_{1} + ⋯ + x_{r} if −x_{1} − … − x_{s} ≤ 0 and h(x_{1}, …, x_{s}) = −x_{r+1} − ⋯ − x_{s} otherwise. In the second case we may make the coordinate change y_{i} = −x_{i} with 1 ≤ i ≤ s and obtain the first case with s − r instead of r. Therefore we consider only the case x_{1}, …, x_{r} > 0, x_{r+1}, …, x_{s} ≤ 0 and x_{1} + … + x_{s} > 0. Then we have to compute the following integral:

∫_{x_{1}=0}^{1} ∫_{x_{2}=0}^{1−x_{1}}⋯ 1−∑_{i=1}^{r−1} x_{i} ; ∫ ; x_{r}=0  0 ; ∫ ; x_{r+1}=− ∑_{i=1}^{r} x_{i} ⋯  0 ; ∫ ; x_{s}=− ∑_{i=1}^{s−1} x_{i} 1 dx_{s} ⋯ dx_{1}

= ∫_{x_{1}=0}^{1} ⋯ 1−∑_{i=1}^{r−1} x_{i} ; ∫ ; x_{r}=0 \frac{(∑_{i=1}^{r} x_{i})^{s−r}}{(s − r)!} dx_{r} ⋯ dx_{1}

= ∫_{x_{1}=0}^{1} ⋯ 1−∑_{i=1}^{r−2} x_{i} ; ∫ ; x_{r−1}=0 \frac{1}{(s − r + 1)!} − \frac{(∑_{i=1}^{r−1} x_{i})^{s−r+1}}{(s − r + 1)!} dx_{r−1} ⋯ dx_{1}

= \frac{1}{(s − r + 1)!(r − 1)!} − ∫_{x_{1}=0}^{1} ⋯ 1−∑_{i=1}^{r−2} x_{i} ; ∫ ; x_{r−1}=0 \frac{(∑_{i=1}^{r−1} x_{i})^{s−r+1}}{(s − r + 1)!} dx_{r−1} ⋯ dx_{1}

= ∑_{j=1}^{r} \frac{(−1)^{j+1}}{(s − r + j)!(r − j)!} = \frac{1}{(s − r)!(r − 1)!s}.

All together we obtain

c_{n,s} = ∑_{r=0}^{s} \frac{2 ( \begin{matrix}s \\ r \end{matrix} )}{(s − r)!(r − 1)!s} = \frac{4^{s}Γ(s + 1/2)}{\sqrt{π}Γ(s + 1)^{2}} = \frac{2Γ(2s)}{s!^{2}Γ(s)} = \frac{(2s)!}{s!^{3}} = \frac{1}{s!}\binom{2s}{s},

where we have used well-known summation formulas for hypergeometric sums and the duplication formula for the Gamma function.

We are left to prove the last statement of the lemma. Since we have |x_{ij}| < 1 for 1 ≤ i ≤ n and 1 ≤ j ≤ s the upper bound is established. For the lower bound we use the fact that all x_{ij} satisfying

∑_{i=1}^{n} ∑_{j=1}^{s} |x_{ij}| < 1

lie in the body that defines c_{n,s}. □

Remark 1. With a little more effort it would be possible to show sharper bounds for c_{n,s}. We abandon to do so because the constant does not effect the asymptotics of u(n; q) which is our main concern in this paper.

Let us consider the result for s = 1, i.e. let us consider the unit group of a real quadratic field, complex cubic field or totally complex quartic field K. Then Theorem 1 yields

u(n; q) = \frac{1}{(n − 1)!} (\frac{ω_{K}(log q)}{log |η|})^{n−1} + o((log q)^{(n−2+ϵ}),

where η is a fundamental unit of K with |η| > 1. Moreover, note that the number of roots of unity ω_{K} = 2 if K is quadratic or a cubic field or a quartic field containing neither i nor \frac{1+i\sqrt{3}}{2}. In particular, for K a real quadratic field we get

u(n; q) = \frac{1}{(n − 1)!} (\frac{2(log q)}{log η})^{n−1} + o((log q)^{(n−2+ϵ}),

which is an improvement to the result obtained in [8].

Moreover, we have an asymptotic formula with explicit main term also for n = 2 and arbitrary s. In the general case we now have the leading term but without an explicit constant.

10 C. Fuchs, R. Tichy and V. Ziegler

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Clemens Fuchs C. Fuchs

ETH Zurich Department of Mathematics Raemistrasse 101 CH-8092 Zurich, Switzerland

e-mail: clemens.fuchs@math.ethz.ch

On the unit sum number problem 11

Robert Tichy R. Tichy

Graz University of Technology Institute for Analysis and Computational Number Theory, Steyrergasse 30, A-8010 Graz, Austria

e-mail: tichy@tugraz.at

Volker Ziegler V. Ziegler

Graz University of Technology Institute for Analysis and Computational Number Theory, Steyrergasse 30, A-8010 Graz, Austria

e-mail: ziegler@finanz.math.tugraz.at