6. Summary

The thesis investigates various aspects of the following problem: Given a set $A=\{a_1 < a_2 < \ldots < a_n\}$ of relatively prime positive integers, which positive integers $K$ can be represented as $K=\sum _{i=1}^n x_ia_i$, where $x_i$ are non-negative integers.

The short introduction is followed by Chapter 2, which plays a double role. On the one hand, we present here all notions necessary for the understanding of the later parts: the greatest non-representable number $G(A)$, the number of non-representable numbers $N(A)$, and the extremal versions of these: $g(n, t)=$max $G(a_1, a_2, \ldots ,a_n)$ and $\nu(n, t)=$max $N(a_1, a_2, \ldots ,a_n)$ where $a_n \le t.$ On the other hand, our presentation, including 15 characteristic problems, is planned to serve also educational purposes. It can be used for teaching talented students; besides the solutions of the above-mentioned problems we include historical remarks, comment on new results and mention also open qestions.

Chapter 3 is devoted to be investigation of $G(A)$ and $g(n, t)$. Our main result is finding the exact value of $g(n, dn+k)$ for two residue classes mod $(d+1)$. This is a generalization of a theorem by ERDŐS and GRAHAM from 1972. Before the proof we present in detail some known upper and lower bounds for these numbers, including a theorem by DIXMIER from 1990, on which the proof of our result is based.

In Chapter 4 we prove that the extremal number $\nu(n,t)$ is obtained when we choose the $n$ largest integers not exceeding $t$. This was a conjecture by ERDŐS and GRAHAM from 1980. We also prove that for infinitely many values of $n$ and $t$, the extremal value $\nu(n,t)$ is achieved also for another set $A$ differing from the set of the greatest $n$ numbers up to $t$.

In the last chapter we deal with the sum $S_k(A)$ of $k^{\rm th}$ powers of the non-representable numbers. First we illustrate, how analysis can be used for determining $S_1(A)$ when $n=2$. Then we discuss some special cases of a general result by RÖDSETH for higher powers. Finally we show a completely elementary method applicable not only to the previous cases, but also for some problems with $n>2.$ This final part can also be considered as a continuation of Chapter 2 in a certain sense, since it can be used as a material for continued training of teachers who know already the basic facts and methods of the topic.

The thesis is based on the following publications of the author: [18] (Chapter 2), [16] (Sections 3.3 - 3.4) and [17] (Section 4.2). (The results of Sections 5.4 - 5.5 are unpublished.)