SOME GENERALIZATIONS OF SUMSETS AND THEIR ASSOCIATED DIRECT AND INVERSE PROBLEMS

Abstract

This thesis deals with the study of sumset and associated problems in additive number theory. The sumset is a fundamental object in additive number theory. In general, the sumset is defined for any additive group G, but the present thesis deals only with the group of integers. Let h ≥ 2 be a positive integer and A1, A2, . . . , Ah be h nonempty finite sets of integers. The sumset of A1, A2, . . . , Ah is defined as A1+A2+· · ·+Ah := {a1+a2+· · ·+ah : ai ∈ Ai for i = 1, . . . ,h}. If Ai = A for i = 1,2, . . . ,h, then the sumset is denoted by hA. This is called h-fold sumset of A. More precisely, if A = {a0,a1, . . . ,ak−1}, then hA := ( k−1 Σ i=0 γiai : γi ∈ {0,1, . . . ,h} for i = 0,1, . . . ,k−1 and k−1 Σ i=0 γi = h ) . If all the summands in h-fold sums are distinct (therefore, h ≤ k), then it is called the restricted h-fold sumset, denoted by h∧A, that is h∧A := ( k−1 Σ i=0 γiai : γi ∈ {0,1} for i = 0,1, . . . ,k−1 and k−1 Σ i=0 γi = h ) . Recently two other variants of these sumsets appeared in the literature; the h-fold signed sumset, h±A, and the restricted h-fold signed sumset, h∧ ±A, which are defined as h±A := ( k−1 Σ i=0 γiai : γi ∈ Z for i = 0,1, . . . ,k−1 and k−1 Σ i=0 |γi| = h ) , and h∧ ±A := ( k−1 Σ i=0 γiai : γi ∈ {−1,0,1} for i = 0,1, . . . ,k−1 and k−1 Σ i=0 |γi| = h ) . Let r and h be positive integers such that 1 ≤ r ≤ h ≤ kr. The generalized h-fold sumset, denoted by h(r)A, is defined by h(r)A := ( k−1 Σ i=0 γiai : γi ∈ {0,1, . . . , r} for i = 0,1, . . . ,k−1 with k−1 Σ i=0 γi = h ) .Therefore, hA and h∧A are particular cases of h(r)A for r = h and r = 1, respectively. This sumset is further generalized to H(r)A, where H is a finite set of positive integers and H(r)A := [ h∈H h(r)A. Mainly, two types of problems are associated with these sumsets; the direct and the inverse problem. In the direct problem, we find the optimal lower bound for the cardinality of the sumset and in the inverse problem we determine the structure of the underlying set when the sumset attains the optimal lower bound. We also consider the extended inverse problem for these sumsets. The characterization of the underlying set(s) for small deviation from the minimum cardinality of the sumset is called an extended inverse problem. In this thesis, we prove some extended inverse theorems for hA for h ≥ 2. Also, we prove some extended inverse theorems for the sumsets 2∧A, 3∧A and 4∧A. Next, we prove some direct and inverse theorems for the sumsets h∧ ±A and H(r)A. We also considered the similar direct and inverse problems for certain subset and subsequence sums of the set A. Let A be a nonempty finite set of integers. The sum of all the elements of a given subset B of A is called a subset sum and it is denoted by s(B). That is, s(B) = Σ b∈B b. The set of all the subset sums of A, denoted by S(A), is the set S(A) = n s(B) : /0 ̸= B ⊆ A o . Also we define, for 1 ≤ α ≤ |A| Sα(A) = n s(B) : B ⊆ A and |B| ≥ α o . The subsequence sum of a given sequence of integers is considered in the same way. Let ⃗r = (r1, r2, . . . , rk) ∈ Nk. Let A = (|a1, .{.z. ,a}1 r1−times ,|a2, .{.z. ,a}2 r2−times , . . . ,|ak, .{.z. ,a}k rk−times ) be a sequence of integers and B = (|a1, .{.z. ,a}1 s1−times ,a|2, .{.z. ,a}2 s2−times , . . . ,|ak, .{.z. ,a}k sk−times ) with 0 ≤ si ≤ ri, for i = 1, . . . ,k, be a subsequence of A. The sum of all terms of the subsequence B is called a subsequence sum, denoted by s(B). That is, s(B) = Σ b∈B b. The set of all subsequence sums of A is the set S(⃗r,A) = {s(B) : B is subsequence of A of length ≥ 1}.For 1 ≤ α ≤ Σki =1 ri, define Sα(⃗r,A) = {s(B) : B is subsequence of A of length ≥ α}. For the sumsets S(A), Sα(A), S(⃗r,A) and Sα(⃗r,A), we prove extended inverse theorem of Freiman (3k−4)-type.