Monatsh MathDOI 10.1007/s00605-013-0574-2

On sums of S-integers of bounded norm

Christopher Frei · Robert Tichy · Volker Ziegler

Received: 28 June 2013 / Accepted: 18 September 2013© Springer-Verlag Wien 2013

Abstract We prove an asymptotic formula for the number of S-integers in a numberfield K that can be represented by a sum of n S-integers of bounded norm.

Keywords Unit equations · S-integers · Algebraic number fields

Mathematics Subject Classification (1991) 11D45 · 11N45

1 Introduction

A (weighted) S-unit equation is an equation over a field K of the form

a1x1 + · · · + an xn = 0,

where ai ∈ K are fixed, xi ∈ � and � is a finitely generated subgroup of K ∗. UsuallyK is a number field and � a group of S-units.

Communicated by J. Schoißengeier.

Dedicated to Wolfgang M. Schmidt on the occasion of his 80th birthday.

C. Frei (B) · R. TichyInstitut für Mathematik A, Technische Universität Graz, Steyrergasse 30,8010 Graz, Austriae-mail: frei@math.tugraz.at

R. Tichye-mail: tichy@tugraz.at

V. ZieglerJohann Radon Institute for Computational and Applied Mathematics (RICAM),Austrian Academy of Sciences, Altenbergerstr. 69, 4040 Linz, Austriae-mail: volker.ziegler@ricam.oeaw.ac.at

123

C. Frei et al.

The study of S-unit equations for their own interest and also in view of applicationshas a long history. Siegel and Mahler [13,18] already considered S-unit equations inorder to prove the finiteness of rational points on certain curves. Nowadays a standardtool to investigate S-unit equations is Wolfgang Schmidt’s famous Subspace Theorem[16] or one of its generalizations due to Evertse, Ferretti or Schlickewei [4,6,14,15].

The theory of S-unit equations was applied, for example, to count the number ofsolutions to norm form equations (see e.g. [5,15]) or to study arithmetic properties ofrecurrence sequences (see e.g. [7,17]).

In this paper we consider applications of S-unit equations to the unit sum numberproblem, i.e. the question which number fields have the property that each algebraicinteger is a sum of units. In the last decade this problem has been studied by severalauthors; we refer to [1] for an overview. We are interested in questions related to aproblem posed by Jarden and Narkiewicz [12, Problem C]:

Problem 1 Let K be an algebraic number field. Obtain an asymptotic formula forthe number Nn(x) of positive integers m ≤ x which are sums of at most n units of thering of integers of K .

Variants of this problem were studied by Filipin, Fuchs, Tichy and Ziegler [8,9].Using a result of Everest [2], they found an asymptotic formula for the number ofnon-associated integers in K of bounded norm that are sums of exactly n units; seealso [3,10]. The aim of this article is to close a gap in [8,9] and to further generalizethe results.

For n, m fixed, we consider integers in K that are sums of exactly n integers in Kof norm at most m. It follows from [11] that not every integer in K can be written assuch a sum. The natural next step is to show a quantitative result in the style of [9].Everest’s theorem allows us to achieve this without much additional effort.

Before we state our main result we fix some notation. Let K be a number field, Sa finite set of places of K containing all Archimedean places, s := |S| − 1 (where |S|denotes the cardinality of the set S), and OK ,S the ring of S-integers of K . We say thattwo S-integers α, β ∈ OK ,S � {0} are associated, α ∼ β, if α/β ∈ O×

K ,S . This is anequivalence relation, and we denote the equivalence class of α by [α]. Let u(n, m; q)

be the number of equivalence classes [α] of nonzero S-integers such that

NS(α) :=∏

v∈S

|α|v ≤ q and α =n∑

i=1

αi ,

where αi ∈ OK ,S � {0} such that NS(αi ) ≤ m and

no subsum of α1 + · · · + αn vanishes. (1)

(The absolute values are normalized by |a|v = |a|[Kv :Qw]w for a ∈ Q, where w is the

place of Q below v and | · |w is the usual w-adic absolute value on Q. By the productformula, NS(α) depends only on [α].)

We write IK ,S(m) for the set of all nonzero principal ideals A of OK ,S of norm[OK ,S : A] ≤ m. Moreover, we define the constant cn,s to be the ns-dimensionalLebesgue measure of

123

On sums of S-integers of bounded norm

{(x11, . . . , xns) ∈ Rns | g(x11, . . . , xns) < 1},

where

g(x11, . . . , xns) =s∑

i=1

max{0, x1i , . . . , xni } + max

{0,−

s∑

i=1

x1i , . . . ,−s∑

i=1

xni

}.

This positive constant is the same as in [9]. It satisfies the inequalities

2ns

(ns)! < cn,s < 2ns,

and its exact values are known for n = 1 or s ≤ 2 [1,9].

Theorem 2 With the above notation the following asymptotic formula holds as q →∞:

u(n, m; q) = |IK ,S(m)|n cn−1,s

n!(

ωK (log q)s

RegK ,S

)n−1

+ O((log q)(n−1)s−1),

where ωK is the number of roots of unity in K and RegK ,S is the S-regulator of K .

We note that the case m = 1 of Theorem 2 is just [9, Theorem 1].

2 Proof of Theorem 2

For each principal ideal A ∈ IK ,S(m), we choose a fixed generator gA ∈ OK ,S , thatis, gAOK ,S = A. Then NS(gA) = [OK ,S : A]. Let G(m) be the set of all gA withA ∈ IK ,S(m). We extend the equivalence relation ∼ to n-tuples in On

K ,S by

(α1, . . . , αn)∼(β1, . . . , βn) :⇔ (α1, . . . , αn) = λ · (β1, . . . , βn)

for some λ ∈ O×K ,S,

and write α = (α1 : · · · : αn) for the equivalence class of (α1, . . . , αn). For α ∈On

K ,S/∼, the expression

NS(α1 + · · · + αn) =∏

v∈S

|α1 + · · · + αn|v ,

is well defined. Each α ∈ OK ,S � {0} with NS(α) ≤ m can be written uniquely asα = g(α)ε, where g(α) is the generator in G(m) of the principal ideal αOK ,S and ε :=α/g(α) ∈ O×

K ,S .Our main tool to prove Theorem 2 is a result due to Everest [2]. We write

Pn−1(O×

K ,S) for the set of equivalence classes (ε1 : · · · : εn) with εi ∈ O×K ,S for

all i ∈ {1, . . . , n}.

123

C. Frei et al.

Theorem 3 (Everest [2]) For fixed c = (c1, . . . , cn) ∈ (K � {0})n, let wc(n; q) bethe number of all (ε1 : · · · : εn) ∈ P

n−1(O×K ,S) such that

NS(c1ε1 + · · · + cnεn) ≤ q

and

no subsum of c1ε1 + · · · + cnεn vanishes. (2)

Then, as q → ∞,

wc(n; q) = cn−1,s

(ωK (log q)s

RegK ,S

)n−1

+ O((log q)(n−1)s−1).

In view of Theorem 2 we are interested in the set

V (n, m; q) := {α ∈ OnK ,S/∼ | NS(α1 + · · · + αn) ≤ q, NS(αi ) ≤ m, (1)}.

Lemma 4 We have, as q → ∞,

|V (n, m; q)| = |IK ,S(m)|ncn−1,s

(ωK (log q)s

RegK ,S

)n−1

+ O((log q)(n−1)s−1).

Proof Condition (1) implies that αi = 0 holds for all i . With ci := g(αi ) and εi :=αi/ci , we can write |V (n, m; q)| as

∑

(c1,...,cn)∈G(m)n

|{(ε1 : · · · : εn) ∈ Pn−1(O×

K ,S) | NS(c1ε1 + · · · + cnεn) ≤ q, (2)}|.

The lemma is an immediate consequence of Theorem 3. �Let us introduce another equivalence relation ∼P on On

K ,S/∼: We say that twoelements α,β ∈ On

K ,S/∼ are equivalent if β arises from α by a permutation of thecoordinates. By [α]P we denote the equivalence class of α ∈ On

K ,S/∼ with respect to∼P . Note that each equivalence class with respect to ∼P has at most n! elements.

Let V ∗(n, m; q) be the set of all α ∈ V (n, m; q) whose equivalence class [α]P hasexactly n! elements.

Lemma 5 We have, as q → ∞,

|V ∗(n, m; q)| = |IK ,S(m)|ncn−1,s

(ωK (log q)s

RegK ,S

)n−1

+ O((log q)(n−1)s−1).

Proof If the equivalence class of α has less than n! elements then there is a permutationπ ∈ Sn, π = id, such that (απ(1) : · · · : απ(n)) = (α1 : · · · : αn), that is,

(απ(1), . . . , απ(n)) = (λα1, . . . , λαn),

123

On sums of S-integers of bounded norm

for some λ ∈ O×K ,S . Assume that, say, π(2) = 1. Since α2 = απn!(2) = λn!α2 and

α2 = 0, we see that λ is a (n!)-th root of unity. We have α1 = λα2 = g(α2)λε2. Inparticular, (1) implies that g(α2)(λ + 1) = 0, so the number of such α in V (n, m; q)

for which there exists such a permutation π is bounded by

∑

(c2,...,cn )∈G(m)n−1

λn!=1,λ=−1

|{(ε2 : · · · : εn) ∈ Pn−2(O×

K ,S) | NS((c2λ + c2)ε2 + · · · + cnεn) ≤ q , (2)}|.

This is � (log q)(n−2)s by Theorem 3. The above argument has to be carried out forall

(n2

)pairs 1 ≤ i < j ≤ n. Thus, the number of equivalence classes [α]P with less

than n! elements is � (log q)(n−2)s , which proves the result. �Clearly, for α ∈ V (n, m; q), the equivalence class [α1 + · · · + αn] depends only

on the equivalence class [α]P . We show that for most of the α ∈ V ∗(n, m; q), thereis no β ∈ V ∗(n, m; q) � [α]P with [β1 + · · · + βn] = [α1 + · · · + αn].Lemma 6 The number of α ∈ V ∗(n, m; q) for which there is β ∈ V ∗(n, m; q)�[α]P

with [α1 + · · · + αn] = [β1 + · · · + βn] is � (log q)(n−2)s as q → ∞.

Proof If there is such a β then we can write α = (α1 : · · · : αn),β = (β1 : · · · : βn),with

α1 + · · · + αn − β1 − · · · − βn = 0.

Since α, β ∈ V ∗(n, m; q), all αi (resp. all βi ) are pairwise distinct. Since β /∈ [α]P ,

{α1, . . . , αn} = {β1, . . . , βn}. (3)

Due to (1) and (3), there exist subsets I, J ⊆ {1, . . . , n} with |I | ≥ 2, such that

∑

i∈I

αi −∑

j∈J

β j = 0 (4)

and no proper subsum of (4) vanishes. Assume that I = {1, . . . , k}, J = {1, . . . , l},for some k ∈ {2, . . . , n}, l ∈ {1, . . . , n}. We write (uniquely) αi = ciεi , βi = diδi ,with ci , di ∈ G(m) and εi , δi ∈ O×

K ,S . Then (4) becomes a non-degenerate S-unitequation

k∑

i=1

ciεi −l∑

j=1

d jδ j = 0, (5)

which has only finitely many solutions (ε1 : · · · : εk : δ1 : · · · : δl) ∈ Pk+l−1(O×

K ,S).Let E = E(c1, . . . , ck, d1, . . . , dl) be the finite set of all representatives of the form

(1, ε2, . . . , εk)

123

C. Frei et al.

that appear in a solution of (5). Then

α = (c1μ : c2ε2μ : · · · : ckεkμ : ck+1εk+1 : · · · : cnεn),

with ci ∈ G(m), (1, ε2, . . . , εk) ∈ E, and μ, εk+1, . . . , εn ∈ O×K ,S . By (1), we have

c1 + c2ε2 + · · · + ckεk = 0. (6)

The number of such α is bounded by the sum over all (c1, . . . , cn) ∈ G(m)n and(1, ε2, . . . , εk) ∈ E with (6) of the number of all

(μ : εk+1 : · · · : εn) ∈ Pn−k(O×

K ,S)

with

NS((c1 + c2ε2 + · · · + ckεk)μ + ck+1εk+1 + · · · + cnεn) ≤ q,

and for which no subsum of (c1 + · · · + ckεk)μ + ck+1εk+1 + · · · + cnεn vanishes.By Theorem 3, this number is � (log q)(n−k)s ≤ (log q)(n−2)s . �

Every equivalence class counted by u(n, m; q) is of the form [α1 + · · · + αn], forsome α = (α1 : · · · : αn) ∈ V (n, m; q). By Lemma 4 and 5, we can restrict ourselvesto equivalence classes coming from α ∈ V ∗(n, m; q) without changing the main term.By Lemma 6, the equivalence class [α]P of each such α is uniquely defined by theclass [α1 + · · · + αn], with � (log q)(n−2)s exceptions. Hence, we obtain the desiredasymptotic formula by counting equivalence classes [α]P with α ∈ V ∗(n, m; q). (Thisargument was neither given in [8] nor [9].)

Acknowledgments V. Z. was supported by the Austrian Science Fund (FWF) under the project P 24801-N26.

References

1. Barroero, F., Frei, C., Tichy, R.F.: Additive unit representations in rings over global fields–a survey.Publ. Math. Debrecen 79(3–4), 291–307 (2011)

2. Everest, G.R.: Counting the values taken by sums of S-units. J. Number. Theory 35(3), 269–286 (1990)3. Everest, G.R., Shparlinski, I.E.: Counting the values taken by algebraic exponential polynomials. Proc.

Amer. Math. Soc. 127(3), 665–675 (1999)4. Evertse, J.-H., Ferretti, R.G.: A further improvement of the quantitative subspace theorem. Ann. of

Math. (2) 177(2), 513–590 (2013)5. Evertse, J.-H., Gyory, K.: The number of families of solutions of decomposable form equations. Acta

Arith. 80(4), 367–394 (1997)6. Evertse, J.-H., Schlickewei, H.P.: A quantitative version of the absolute subspace theorem. J. Reine

Angew. Math. 548, 21–127 (2002)7. Evertse, J.-H., Schlickewei, H.P., Schmidt, W.M.: Linear equations in variables which lie in a multi-

plicative group. Ann. of Math. (2) 155(3), 807–836 (2002)8. Filipin, A., Tichy, R.F., Ziegler, V.: On the quantitative unit sum number problem–an application of

the subspace theorem. Acta Arith. 133(4), 297–308 (2008)9. Fuchs, C., Tichy, R.F., Ziegler, V.: On quantitative aspects of the unit sum number problem. Arch.

Math. 93, 259–268 (2009)

123

On sums of S-integers of bounded norm

10. Gyory, K., Mignotte, M., Shorey, T.N.: On some arithmetical properties of weighted sums of S-units.Math. Pannon. 1(2), 25–43 (1990)

11. Hajdu, L.: Arithmetic progressions in linear combinations of S-units. Period. Math. Hung. 54(2),175–181 (2007)

12. Jarden, M., Narkiewicz, W.: On sums of units. Monatsh. Math. 150(4), 327–332 (2007)13. Mahler, K.: Zur Approximation algebraischer Zahlen (I). Math. Ann. 107, 691–730 (1933)14. Schlickewei, H.P.: Linearformen mit algebraischen Koeffizienten. Manuscripta Math. 18(2), 147–185

(1976)15. Schlickewei, H.P.: The p-adic Thue-Siegel-Roth-Schmidt theorem. Arch. Math. (Basel) 29(3), 267–270

(1977)16. Schmidt, W.M.: Linear forms with algebraic coefficients I. J. Number Theory 3, 253–277 (1971)17. Schmidt, W.M.: Linear recurrence sequences. In: Diophantine approximation (Cetraro, 2000), Lecture

Notes in Math., vol. 1819, pp. 171–247. Springer, Berlin (2003)18. Siegel, C.L.: Über einige Anwendungen Diophantischer Approximationen. Abh. Preuss. Akad. Wiss.

Phys. Math. Kl. 41–69 (1929)

123