On sumsets of nonbases of maximum size (Q6612511)

Let \(G\) be a finite additive abelian group. A nonempty subset \(A\) in \(G\) is called a basis of order \(h\) if \(hA = G\); that is, every element of \(G\) can be written as a sum of \(h\) elements of \(A\). If \(A\) is not a basis of order \(h\), it is called a non-basis of order \(h\). The \(h\)-critical number \(\chi(G, h)\) of \(G\) is defined as the smallest positive integer \(m\) for which all \(m\)-subsets of \(G\) are \(h\)-bases; that is\N\[\N\chi(G, h) = \min\{m: A \subset G, |A| \ge m \Rightarrow hA = G\}.\N\]\NThe \(h\)-critical number of a group is a well-defined and well-studied object in additive combinatorics. It is natural to ask what are the possible sizes of \(hA\) if \(A\) is a non-basis of order \(h\) of maximum size in \(G\)? Namely, the set\N\[\NS(G,h) = \{|hA|: A \subset G, |A| = \chi(G,h)-1, hA \ne G \}.\N\]\N\NIn this paper, the authors completely answer this question for \(h = 2\) and \(h = 3\). They prove the following theorems for \(h = 2\) and \(h = 3\), respectively.\N\NTheorem 1.1. Let \(G\) be an abelian group of order \(n\).\N\N1. When \(n\) is even, the maximum size of a 2-incomplete subset of \(G\) is \(n/2\), and the elements of \(S(G, 2)\) are of the form \(n - n/d\) where \(d\) is some even divisor of \(n\); in fact all such integers are possible, with the exception that \(3n/4\) arises only when the exponent of \(G\) is divisible by 4. \N\N2. When \(n\) is odd, the maximum size of 2-incomplete subsets of \(G\) is \((n-1)/2\); furthermore, when \(G\) is of order 3, 5, or is noncyclic and of order 9, then \(S(G, 2) = \{n-2\}\), and for all other groups of odd order we have \(S(G, 2) = \{n - 2, n - 1\}\).\N\NTheorem 1.2. Let \(G\) be an abelian group of order \(n\). \N\N1. When \(n\) has prime divisors congruent to \(2 \mod 3\), and \(p\) is the smallest such prime, the maximum size of a 3-incomplete subset is \((p + 1)n/(3p)\), and we have \(S(G, 3) = \{n - n/p\}\).\N\N2. When \(n\) is divisible by 3 but has no divisors congruent to \(2 \mod 3\), then the maximum size of a 3-incomplete subset is \(n/3\), and the elements of \(S(G, 3)\) are of the form \(n-n/d\) or \(n-2n/d\) where \(d\) is some divisor of \(n\) that is divisible by 3; furthermore, all such integers are possible, with the exceptions of \(2n/3\) and \(n - 2n/d\) when the highest power of 3 that divides \(d\) is more than the highest power of 3 that divides the exponent of \(G\).\N\N3. In the case when all divisors of \(n\) are congruent to \(1 \mod 3\), then the maximum size of a 3-incomplete subset is \((n - 1)/3\), and \(S(G, 3) = \{n - 3, n - 1\}\), unless \(G\) is an elementary abelian 7-group, in which case \(S(G, 3) = \{n - 3\}\).\N\NUsing Kneser's theorem as the main tool, the authors prove Theorems 1.1 and 1.2 using elementary methods.