Weak asymptotics in the 3-dim Frobenius problem. (English) Zbl 1216.11092
In this paper the author considers a few related conjectures of Arnold on the properties of numerical semigroups with exactly three minimal generators. Recall that a numerical semigroup is an additive submonoid of \(\mathbb{N}_0\) with finite complement in \(\mathbb{N}_0\). The largest element of the complement is called the Frobenius number of the semigroup. The Frobenius number plus one is called the conductor.
One of Arnold’s conjectures discussed in this paper, Problem #1999-8 in the book [Arnold’s problems. Berlin: Springer (2004; Zbl 1051.00002)], asks for the average value of the conductor as we take a fixed numerical semigroup with generators \(d_1,d_2, d_3\) and consider the semigroups with generators \(Nd_1 + j_1, N d_2 + j_2, N d_3 + j_3\) where each \(j_i \in [-r,r]\) and \(r\) and \(N\) go to infinity with \(r\) small relative to \(N\). A related conjecture asks for statistics for the number of elements less than the Frobenius number contained in a semigroup under a similar kind of averaging. This type of averaging is known as the study of “weak asymptotics at typical large vectors”.
The author studies these problems in detail and refutes Arnold’s conjectures in the case of semigroups with three minimal generators. The approach builds on the author’s earlier work [Funct. Anal. Other Math. 1, No. 2, 119–157 (2006; Zbl 1194.20058)].
I believe that this paper would have benefitted greatly from more careful editing. The exposition is sometimes unclear and contains several mistakes in the use of English. Some of the proofs, particularly in the paper’s final section, could also be improved.
One of Arnold’s conjectures discussed in this paper, Problem #1999-8 in the book [Arnold’s problems. Berlin: Springer (2004; Zbl 1051.00002)], asks for the average value of the conductor as we take a fixed numerical semigroup with generators \(d_1,d_2, d_3\) and consider the semigroups with generators \(Nd_1 + j_1, N d_2 + j_2, N d_3 + j_3\) where each \(j_i \in [-r,r]\) and \(r\) and \(N\) go to infinity with \(r\) small relative to \(N\). A related conjecture asks for statistics for the number of elements less than the Frobenius number contained in a semigroup under a similar kind of averaging. This type of averaging is known as the study of “weak asymptotics at typical large vectors”.
The author studies these problems in detail and refutes Arnold’s conjectures in the case of semigroups with three minimal generators. The approach builds on the author’s earlier work [Funct. Anal. Other Math. 1, No. 2, 119–157 (2006; Zbl 1194.20058)].
I believe that this paper would have benefitted greatly from more careful editing. The exposition is sometimes unclear and contains several mistakes in the use of English. Some of the proofs, particularly in the paper’s final section, could also be improved.
Reviewer: Nathan Kaplan (Cambridge)
MSC:
| 11P21 | Lattice points in specified regions |
| 11D07 | The Frobenius problem |
| 11N56 | Rate of growth of arithmetic functions |
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