Thin Bases of Order Two

AbstractA set A⊆N0 is called a basis of order two if A+A≔{a+a′∣a, a′∈A}=N0. If n∈N then A(n) denotes the number of a∈A with 1⩽a⩽n. In this paper bases A, B, C of order two are given such thatlimA(n)n=253,limB(n)n=72andlimC(n)n=101653.

Linear Diophantine Problems

The Frobenius number g(A k ) Let A k \({A_k} = \{ {a_1},...,{a_k}\}\subset\) IN with gcd(A k ) = 1, n\( \in I{N_0}.\) If $$n = \sum\limits_{i = 1}^k {{x_i}{a_i},{x_i}}\in I{N_0}$$ (1) we call this a representation or a g-representation of n by Ak (in order to distinguish between several types of representations that will be considered in the sequel). Then the Frobenius number g(A k ) is the greatest integer with no g-representation.

Extremal Frobenius numbers in a class of sets

For given $ A_k=\{ a_1,\ldots ,a_k \}, a_1 \le \ldots \le a_k $ coprime the Frobenius number $ {g}(A_k) $ is defined as the greatest integer ${g}$ with no representation¶¶ ${g}=\sum \limits ^k_{i=1}\,x_i\,a_i,\;x_i\in {\Bbb N}_0 $ . ¶¶A class $ {\bf A}^*_k $ is given, such that ¶¶ $ {\overline {g}}^*(k,y):= \max \{ {g}(A_k)|A_k\in {\bf A}^*_k,\, a_k\le y \} $ ¶¶has the same asymptotic behaviour as the general function¶¶ $ {\overline {g}}(k,y):= \max \{ {g}(A_k)| a_k\le y \}\, {\rm for} \, y\to \infty $ .¶¶ Furthermore, ¶¶ $ {\underline {g}}^*(k,x):= \min \{ {g}(A_k)|A_k\in {\bf A}^*_k,\, a_1\ge x \} $ ¶¶is shown to have the same order of magnitude as the general function¶¶ $ {\underline {g}…

Remark on linear forms