An Estimate for Frobenius’ Diophantine Problem in Three Dimensions

Jan-Christoph Schlage-Puchta Mathematisches Institut Eckerstr. 1 79104 Freiburg Germany

                jcsp@mathematik.uni-freiburg.de

Abstract

We give upper and lower bounds for the largest integer not representable as a positive linear combination of three given integers, disproving an upper bound conjectured by Beck, Einstein and Zacks.

1 Introduction

Let a, b, c be integers. The Frobenius problem is to determine the greatest integer n which cannot be represented as a positive linear combination of a, b, c; this integer will henceforth be denoted by f(a, b, c). Of course, if a, b, c have a common divisor m ≥ 2, an integer not divisible by m cannot be represented as an integral linear combination of a, b, and c, we will therefore suppose that gcd(a, b, c) = 1.

Beck, Einstein and Zacks[1] conjectured that, apart from certain explicit families of exceptions, we have f(a, b, c) ≤ (abc)^{5/8}. They supported their conjecture by an impressive amount of numerical data and stated that counterexamples are not to be expected unless a, b, c are close to some arithmetic progression. However, in this note we will show that their conjecture fails for somewhat generic examples, while it holds true for the vast majority of triples. We do so by using diophantine approximations inspired by the circle method. The idea to use such an approach is already present in the work of Beck, Diaz and Robins[2]; however, it has not yet been fully exploited.

2 The Main Result

Define N_{a}(b, c) to be the least integer n such that for every integer x the congruence x ≡ νa + μb (mod a) is solvable with 0 ≤ ν, μ ≤ n. The basic relations between the functions f and N are summarized as follows.

Proposition 1. Let a, b, c be positive integers satisfying gcd(a, b, c) = 1.

(i) We have

min(b, c)N_{a}(b, c) ≤ f(a, b, c) ≤ (b + c)N_{a}(b, c).

(ii) Suppose that gcd(a, b) = 1. Let \overline{b} denote the modular inverse of b, modulo a. Then we have N_{a}(b, c) = N_{a}(1, c\overline{b}).

(iii) Suppose that gcd(p, q) = 1 and

|\{\frac{c\overline{b}}{a}\} − \frac{p}{q}| = δ ≤ \frac{1}{q^{2}} .

Then

min (\frac{a}{q}, \frac{1}{δq} ) ≪ N_{a}(b, c) ≪ \frac{a}{q} + q.

Proof. (i) In every residue class modulo a there exists an element x which can be represented as x = bν + cμ with 0 ≤ ν, μ ≤ N_{a}(b, c), in particular, x ≤ (b + c)N_{a}(b, c). Hence, every integer n ≥ (b + c)N_{a}(b, c) can be written as n = aα + bν + cμ with α ≥ 0 and 0 ≤ ν, μ ≤ N_{a}(b, c). Thus, f(a, b, c) ≤ (b + c)N_{a}(b, c). Conversely, there exists an integer x such that the congruence x ≡ νa + μb (mod a) is unsolvable with 0 ≤ ν, μ ≤ N_{a}(b, c) − 1; that is, every element in this class which is representable by a, b, c can only be represented using N_{a}(b, c) summands b or c, and is therefore of size at least min(b, c)N_{a}(b, c). That is, f(a, b, c) ≥ min(b, c)N_{a}(b, c).

(ii) Our claim follows immediately from the fact that x ≡ bν + cμ (mod a) is equivalent to x\overline{b} ≡ ν + c\overline{b}μ (mod a).

(iii) Define the integers 0 = x_{0} < x_{1} < ⋯ < x_{q−1} < a by the relation {x_{0}, …, x_{q−1}} = \{0, c\overline{b} mod a, 2c\overline{b} mod a, …, (q − 1)c\overline{b} mod a\}. Then we have

max(x_{i+1} − x_{i}) ≤ \frac{a}{q} + (q − 1)δ ≤ \frac{a}{q} + q;

hence, every residue class x modulo a can be written as ν +μc\overline{b} with ν ≤ \frac{a}{q} +q and μ ≤ q −1. Hence N_{a}(1, c\overline{b}) ≤ \frac{a}{q} + q. Together with (ii), the upper bound follows. For the lower bound

suppose, without loss of generality, that \{\frac{c\overline{b}}{a}\} > \frac{p}{q}, and consider x = ⌊\frac{a}{q} − 1⌋. We claim that

x cannot be represented with fewer than \frac{1}{2} min (\frac{a}{q}, \frac{1}{δq}) summands modulo a.

First suppose that δ < \frac{1}{a}. Then 1 ≤ qc\overline{b} mod a ≤ q − 1, and no multiple μc\overline{b} mod a with q ∤ μ, μ ≤ \frac{a}{q} falls in the range [1, x]. Hence, for all pairs ν, μ with 1 ≤ ν, μ ≤ \frac{a}{2q}, such that ν + μc\overline{b} mod a ∈ [1, x] we have q|μ and

ν + μc\overline{b} mod a ≤ ν + \frac{q − 1}{q}μ < ⌊\frac{a}{q}⌋,

which implies our claim in this case.

If δ > \frac{1}{a}, and μ < \frac{1}{2δq}, μc\overline{b} mod a either does not fall into the range [1, x], or satisfies μc\overline{b} mod a ≤ δμ, which implies the lower bound in this case as well.

From this result we draw the following conclusion:

Theorem 1. For each integer a there are \frac{φ(a)}{2} pairs (b, c) such that a < b < c < 2a, gcd(a, b) = gcd(a, c) = 1, and \frac{a^{2}}{2} ≤ f(a, b, c) ≤ 2a^{2}. Moreover, for ϵ > 0, α, β ∈ [0, 1] and a > a_{0}(ϵ), there exist b ∈ [(1 + α)a, (1 + α + ϵ)a] and c ∈ [(1 + β)a, (1 + β + ϵ)a], such that f(a, b, c) ≥ \frac{a^{2}}{2} . On the other hand, the number of pairs (b, c) such that a < b < c < 2a, (a, b, c) = 1 and f(a, b, c) > a^{3/2+δ} is bounded above by O(a^{2−2δ}).

Proof. Choose b ∈ [a, 2a] subject to the condition gcd(a, b) = 1, then choose c ∈ [a, 2a] such that bc ≡ 1 (mod a). The proposition implies that N_{a}(b, c) ≥ \frac{a}{2}, whereas the upper bound follows from f(a, b, c) ≤ f(a, b) ≤ 2a^{2}. Since there are φ(a) choices for b, and at least half of them satisfy b < \overline{b}, our first claim follows. For the second claim it suffices to prove that for every α, β ∈ [0, 1] there exist integers b ∈ [(1 + α)a, (1 + α + ϵ)a] and c ∈ [(1+β)a, (1+β +ϵ)a], such that bc ≡ 1 (mod a). However, this follows immediately from Weil’s estimate for Kloosterman sums[4] together with the Erdős-Turán-Koksma inequality (cf., e.g., [3, p. 116]). Finally, let δ > 0 be fixed, and set Q = a^{1/2+δ}. For every real number α ∈ [0, 1] there exists some q ≤ Q, such that |α − \frac{p}{q}| ≤ \frac{1}{qQ}. Denote by 𝔐 the set of all α ∈ [0, 1], such that |α − \frac{p}{q}| < \frac{1}{q^{2}} does not hold for any q ∈ [a^{1/2−δ}, a^{1/2+δ}]. Then 𝔐 consists of a^{1−2δ} intervals of total measure bounded above by

∑_{q≤a^{1/2−δ}} \frac{φ(q)}{qQ} ≤ a^{−2δ},

thus, there are O(a^{1−2δ}) integers ν, such that N_{a}(1, ν) ≥ a^{1/2+δ}, and our claim follows.

References

[1] M. Beck, D. Einstein, S. Zacks, Some experimental results on the Frobenius problem, Experiment. Math. 12 (2003), 263–269.

[2] M. Beck, R. Diaz, S. Robins, The Frobenius problem, rational polytopes, and Fourier-Dedekind sums, J. Number Theory 96 (2002), 1–21.

[3] L. Kuipers, H. Niederreiter, Uniform distribution of Sequences, Pure and Applied Mathematics. Wiley, New York, London, Sydney, Toronto, 1974

[4] A. Weil, On some exponential sums, Proc. Nat. Acad. Sci. U. S. A. 34 (1948), 204–207.

2000 Mathematics Subject Classification: Primary 11D04.

Keywords: Frobenius problem, linear diophantine equations, diophantine approximation.

Received January 26 2005; revised version received February 7 2005. Published in Journal of Integer Sequences, March 1 2005.

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