Additive completion and disjoint translations
\(A\) set \({B}\) is called an additive complement of the sequence \({A} \subset [0,N]\) if \(A+B: =\{a+b:a \in A;b\in B\} \supset\{0,1, \dots, N\}\). Define the covering number \({L}\) of \({A}\) as \(L=\min \{|B|: B\subset \mathbb{Z},\;A+B \supset \{0,1, \dots, N\}\}\). So the set \({A}\) is well-defined in this sense if \(L \cdot |A|- (N+1)\) is ``small''. The authors introduce the fractional covering number as \(\Lambda = \min \sum_{k = -N}^N \lambda_k\), where \(\lambda_k\geq 0\) and \(\sum_{a\in A} \lambda_{n-a} \geq 1\) for all \(0\leq n\leq N\). Clearly \(\Lambda\leq L\). They give a generalization of a theorem of the first author proving: Let \(A=\{P(n): n\in\mathbb{N},\;0\leq P(n)\leq N\}\), where \(P(n)\) is a polynomial of degree \(d>1\), with integral coefficients and positive leading coefficient. For a fixed polynomial and \(N\to\infty\) \[ \Lambda = \bigl(1+o(1)\bigr) s(1/d) {N\over|A|}, \quad \text{where} \quad s(t)= {\sin\pi t\over \pi t(1-t)}. \] They also introduce the notion of disjoint translation number \(D\) as \[ D=\sup \bigl\{|B|:B \subset [0,N\cap \mathbb{Z},\;A+ b_1 \cap A+b_2= \emptyset\;\forall b_1,b_2 \in B,\;b_1 \neq b_2\bigr\}. \] The fractional analog of \(D\) is the following: \[ \Delta= \max\sum^N_{k=0} \delta_k,\;\delta_k\geq 0 \quad \text{and} \quad \sum_{a \in A,\;n-N \leq a\leq N} \delta_{n-a} \leq 1 \] for all \(0\leq n\leq 2N\). They prove \(D\leq \Delta = \Lambda \leq L\). So the above mentioned result is an estimation for \(\Delta\) (and so for \(D)\). Finally they determine the asymptotic behaviour of \(\Lambda\) for a wider class of sequences. They prove: if for all \(t\in[0,1]\), \(\lim_{N\to\infty} A(tN)/A(N) = t^\alpha\) for some \(\alpha\in(0,1)\), then \[ \Lambda = \bigl(1+o(1) \bigr) s(\alpha) {N\over A(N)}, \quad \text{where} \quad A(N)= \sum_{a_i\leq N}1. \]
scientific article; zbMATH DE number 878104
scientific article