Generalized H-fold sumset and Subsequence sum

Let $A$ and $H$ be nonempty finite sets of integers and positive integers, respectively. The generalized $H$-fold sumset, denoted by $H^{(r)}A$, is the union of the sumsets $h^{(r)}A$ for $h\in H$ where, the sumset $h^{(r)}A$ is the set of all integers that can be represented as a sum of $h$ elements from $A$ with no summand in the representation appearing more than $r$ times. In this paper, we find the optimal lower bound for the cardinality of $H^{(r)}A$, i.e., for $|H^{(r)}A|$ and the structure of the underlying sets $A$ and $H$ when $|H^{(r)}A|$ is equal to the optimal lower bound in the cases $A$ contains only positive integers and $A$ contains only nonnegative integers. This generalizes recent results of Bhanja. Furthermore, with a particular set $H$, since $H^{(r)}A$ generalizes subsequence sum and hence subset sum, we get several results of subsequence sums and subset sums as special cases.


Introduction
Let N be the set of positive integers.Let A = {a 1 , . . ., a k } be a nonempty finite set of integers and h be a positive integer.The h-fold sumset, denoted by hA, and the restricted h-fold sumset, denoted by h ∧ A of A, are defined, respectively, by Mistri and Pandey [6] generalized hA and h ∧ A, into the generalized h-fold sumset, denoted by h (r) A, as follows: Let r be a positive integer such that 1 ≤ r ≤ h.The generalized h-fold sumset h (r) A, is defined by So, the generalized h-fold sumset h (r) A is the set of all sums of h elements of A, in which every summand can repeat at most r times.Therefore, hA and h ∧ A are particular cases of h (r) A for r = h and r = 1, respectively.
For a finite set H of positive integers, Bajnok [1] introduced the sumset and the restricted sumset In a recent article, Bhanja and Pandey [5] considered a generalization of HA and H ∧ A, the generalized H-fold sumset, denoted by H (r) A, defined by Observed that, if r ≥ max(H), then H (r) A = HA and if r = 1, then H (r) A = H ∧ A. The sumset H (r) A becomes more important as it also generalizes subset sums and subsequence sums.
1.1.Subset sum and Subsequence sum.Let A be a finite set of integers.The sum of all the elements of a given subset B of A is called subset sum and it is denoted by s(B).That is, The set of all nonempty subset sum of A, denoted by (A), that is Also we define, for 1 ≤ α ≤ k α (A) = s(B) : ∅ = B ⊆ A and |B| ≥ α .
Similarly, we define subsequence sum of a given sequence of integers.Let A = {a 1 , a 2 , . . ., a k } be a set of k integers and r be a positive integer, with a 1 < a 2 < • • • < a k .Then we define a sequence associated with A as Given any subsequence B of A, the sum of all terms of the subsequence B is called the subsequence sum, is denoted by s(B) and we write The set of all subsequence sums of a given sequence A is the set (A) = {s(B) : B is subsequence of A of length ≥ 1} .
For 1 ≤ α ≤ kr, define Note that, we can write With suitable sets H, we can express (A), α (A), (A) and α (A) in terms of H ∧ A and Let A = {a 1 , a 2 , . . ., a k } be a nonempty set of integers with a 1 < a 2 < • • • < a k .For an integer c, we write c * A = {ca : a ∈ A} and for integers a and b with a < b, we write [a, b] = {a, a + 1, . . ., b}.For a nonempty set S = {s 1 , s 2 , . . ., s n−1 , s n }, we let max(S), min(S), max − (S), min + (S) be the largest, smallest, second largest and second smallest elements of S, respectively.For a given real number x, ⌊x⌋ and ⌈x⌉ denote, floor function and ceiling function of x, respectively.We assume t i=1 f (i) = 0 if t < 1.Two standard problems associated with a sumset in additive number theory are to find best possible lower bound for the cardinality of sumset when the set A is known (called the direct problem) and to find the structure of the underlying set A when the size of the sumset attains its lower bound (called the inverse problem).These two types of problems have been solved for the sumsets in various types of groups.We have several classical results on sumsets for the case when A is a subset of group of integers, (see [1], [4], [6], [8], [9], [10], [11]), and for subsequence sums and subset one may refer to [2], [3], [5], [7].We mention now, some of these results that are applied in this paper.Mistri and Pandey [6] generalized above results as follows: Theorem 1.3.[6, Theorem 2.1] Let A be a nonempty finite set of k integers.Let r and h be integers such that 1 ≤ r ≤ h ≤ kr.Set m = ⌊h/r⌋.Then This lower bound is best possible.
then A is an arithmetic progression.
Further generalization of h (r) A was considered in [6] for which the direct and inverse results were proved by Yang and Chen [11].Direct results for h (r) A when A is a subset of the group of residual classes modulo a prime and A is a subset of a finite cyclic group were given, respectively, by Monopoli [8] and Bhanja [4].
The direct and inverse theorems for the sumsets HA and H ∧ A proved by Bhanja [2] are the following: Theorem 1.5.[2, Theorem 3] Let A be a set of k positive integers.Let H be a set of t positive integers with max(H) = h t .Then This lower bound is optimal.
Theorem 1.6.[2, Theorem 5] Let A be a set of k ≥ 2 positive integers and H be a set of t ≥ 2 positive integers with max(H) = h t .If then H is an arithmetic progression with common difference d and A is an arithmetic progression with common difference d * min(A).
Theorem 1.7.[2, Theorem 6, Corollary 7] Let A be a set of k nonnegative integers and H = {h 1 , h 2 , . . ., h t } be a set of positive integers with The lower bounds are optimal.
Theorem 1.8.[2, Theorem 9, Corollary 10] Let A be a set of k nonnegative integers.Let H = {h 1 , h 2 , . . ., h t } be a set of positive integers with and In this paper, we prove similar direct and inverse results for the sumset H (r) A when A is a finite nonempty set of positive integers.In sections 2 and 3, we prove our main theorems, Theorem 2.1 and Theorem 3.1, the direct and inverse theorems for sumset H (r) A, when A is a finite set of positive integers.Consequentaly we prove direct and inverse theorems when A contains nonnegative integers with 0 ∈ A.
To state our main results we need some notation that are used throughout the paper.Let H = {h 1 , h 2 , . . ., h t } be a set of positive integers with 0 = h giving H (r) A = HA.So we assume that r ≤ h t .There always exists a unique positive integer l such that h l−1 < r ≤ h l , where 1 ≤ l ≤ t.For i = 1, 2, . . ., t, let h i = m i r + ǫ i , where 0 ≤ ǫ i ≤ r − 1.For given set H of positive integers and set of integers A with |H|= t and |A|= k, let Note that, if 0 ≤ i ≤ l − 1, then m i = 0 and ǫ i = h i .So, we can also write Also, let {0} (r) A = {0}.

Direct Theorems
Theorem 2.1.Let A be a nonempty finite set of k ≥ 3 positive integers.Let r be a positive integer and H be a set of t ≥ 2 positive integers with Then This lower bound is best possible.
Since l is a positive integer satisfying h l−1 < r ≤ h l , we have m i = 0 and

and by Theorem 1.1, we have |R
then we define T i for every possible values of ǫ i−1 and ǫ i , and consequently find Let T i = ∅ in this case.Then by Theorem 1.3, we have Then, we have max Then by Theorem 1.3 and (3), we have and for j = 0, 1, . . ., r − ǫ i , Then, we have max ) Then by Theorem 1.3 and (4), we have For j = 0, . . ., r − ǫ i − 1 and q = 1, . . ., m i − m i−1 − 1, define Then by Theorem 1.3 and (5), we have Then by Theorem 1.3 and (6), we have and It is easy to see that U 0 i,r−ǫ i = max(h Then by Theorem 1.3 and (7), we have Then by Theorem 1.3 and (8), we have Define also It is easy to see that T 0 i,r−ǫ i−1 = max(h Then by Theorem 1.3 and (9), we have Hence This proves (2).Next, we show that this bound is best possible.
On the other hand, we have by ( 2), This completes the proof of Theorem 2.1.
This lower bound is best possible, as that can be verified with where m 1 = ⌊h 1 /r⌋.To check, this bound is best possible, we take Corollary 2.1.Let A be a nonempty finite set of k ≥ 4 nonnegative integers with 0 ∈ A. Let r be a positive integer and H be a set of t ≥ 2 positive integers with 1 ≤ r ≤ max(H) ≤ (k − 2)r − 1.Let m = ⌈min(H)/r⌉ and m 1 = ⌊min(H)/r⌋.Then This lower bound is best possible. Proof.
. Hence by Theorem 1.3 and Theorem 2.1, we have This proves the Corollary.To check optimallity of the bound, take From (10), we have Remark 2.3.Following the notation from Corollary 2.1.
This lower bound is best possible, as that can be verified with Therefore where m 1 = ⌊h 1 /r⌋.To check, this bound is best possible, we take and hence H (r) A = (2k − 3)r + 1.
(a) For r = max(H) = h t , Theorem 1.5 follows from Theorem 2.1 as a consequence.(b) For r = 1, Theorem 1.7 follows from Remark 2.2 and Remark 2.3 as a consequence.

Inverse problem
This section deals with the inverse theorems associated with the sumset H (r) A. In this section, we charaterize the sets A and H, when the cardinality of H (r) A is equal to its optimal lower bound.There are some cases in which either A or H or both may not be arithmetic progression but size of H (r) A is equal to the optimal lower bound (called extremal set).See some extremal sets in [6, Section 3] and [3, Section 2.2].Here we give some more example of extremal sets.
(1) Let A be a set of k (≥ 3) integers and r be a positive integer.If H = {1, rk} or } with H ⊆ {1, 2, 3} and r = 1.Then the sets A are extremal sets.
We now present the main inverse results associated with H (r) A.
Theorem 3.1.Let r ≥ 1 be a positive integer, A be a nonempty finite set of k ≥ 6 positive integers and H be a set of t ≥ 2 positive integers with then H is an arithmetic progression with common difference d ≤ r and A is an arithmetic progression with common difference d * min(A).
For i = 1, . . ., t, let h i = m i r + ǫ i , where 0 ≤ ǫ i ≤ r − 1.Let l be a positive integer such that h l−1 < r ≤ h l , where 1 ≤ l ≤ t.Since |H (r) A| is equal to its lower bound given in (2), we have, from the proof of Theorem 2.1 that, |H (r) A|= t i=1 |R i |.This implies that then by Theorem 1.4, the set A is an arithmetic progression.Let h 1 = 1 and h 2 > 2. Then we have Therefore |S 2 | is minimum and hence A 2 = {a 1 , a 2 , . . ., a k−1 } is an arithmetic progression.Now we show that Let m 2 = k − 2 and ǫ 2 = 0. Then This implies that ra 1 + . . .
and by Theorem 1.3 Let y be an element of h (r) 2 A, which is different from min(h This gives H (r) A > t i=1 |R i |, which is not possible.Therefore y ∈ R 2 .This gives that h and so by Theorem 1.4,A is an arithmetic progression. Let , where d 1 is the common difference of A. We show that H is an arithmetic progression with common difference d and

This implies that min{(h
a 1 for some s.(11) Consider the following cases: Hence, (h i+1 −h i )a 1 = a 2 −a 1 = d 1 for each i = 1, . . ., t−1.So H is an arithmetic progresion with common difference d ≤ r and d 1 = da 1 .This completes the proof.and where A ′ = {a 1 , a 2 , . . ., a k−1 } and B = {0, a 1 , . . ., a m } with m = ⌈h 1 /r⌉.Then by Theorem 3.1, H is an arithmetic progression with common difference d ≤ r and A ′ is an arithmetic progression with common difference d * min(A ′ ).Now, we show that d = 1, if h 1 > 1.To show d = 1, it is sufficient to prove that common difference of arithmetic progression A is a 1 .If r = 1, then d = 1.Assume r ≥ 2. Now, define R i = S i ∪ T i for the set A ′ as it was defined in Theorem 2.1.So therefore Theorem 1.4 implies that B is an arithmetic progression with common difference a 1 and we know that A ′ is also an arithmetic progression.Hence A = B ∪ A ′ is an arithmetic progression with common difference a 1 .♣ If h 1 = m 1 r, then m 1 = m, and if h 1 < r, then m 1 = 0 and m = 1.Since h 1 A, so from (14), we have Corollary 3.5.Let r ≥ 1 and t > t 0 ≥ 2 be integers.Let A be a nonempty finite set of k ≥ 7 nonnegative integers with 0 ∈ A and H = {h 1 , h 2 , . . ., h t } be a set of t positive integers with h 1 < h 2 < • • • < h t 0 −1 ≤ (k − 2)r − 1 < h t 0 < • • • < h t < (k − 1)r.If (t 0 , h 1 ) = (2, 1) and H (r) A ≥ m 1 r(m − m 1 + 1) + (min(H) − m 1 r)(m − 2m 1 ) + L H (r) (A \ {0}) + t − t 0 + 2, then H is an arithmetic progression with common difference d ≤ r and A is an arithmetic progression with common difference d * min(A \ {0}).Moreover, if min(H) > 1, then d = 1.

This gives
Corollary 3.6.Let r ≥ 1 and t ≥ 2 be integers.Let A be a nonempty finite set of k ≥ 7 nonnegative integers with 0 ∈ A and H = {h 1 , h 2 , . . ., h t } be a set of t positive integers with

Conclusions
In §1.1, we have already discussed the relation between generalized H-fold sumset and subsequence sum.Choosing a particular H we get some results of subsequence sum.The above lower bounds are best possible.
This lower bound is best possible.Furthermore, if |hA| attains this lower bound with h ≥ 2, then A is an arithmetic progression.Theorem 1.2.[9, 10, Theorem 1, Theorem 2] Let A be a nonempty finite set of integers, and let 1 ≤ h ≤ |A|.Then |h ∧ A| ≥ h |A| − h 2 + 1.This lower bound is best possible.Furthermore, if |h ∧ A| attains this lower bound with |A| ≥ 5 and 2 ≤ h ≤ |A|−2, then A is an arithmetic progression.