On the minimum size of subset and subsequence sums in integers

Let $\mathcal{A}$ be a sequence of $rk$ terms which is made up of $k$ distinct integers each appearing exactly $r$ times in $\mathcal{A}$. The sum of all terms of a subsequence of $\mathcal{A}$ is called a subsequence sum of $\mathcal{A}$. For a nonnegative integer $\alpha \leq rk$, let $\Sigma_{\alpha} (\mathcal{A})$ be the set of all subsequence sums of $\mathcal{A}$ that correspond to the subsequences of length $\alpha$ or more. When $r=1$, we call the subsequence sums as subset sums and we write $\Sigma_{\alpha} (A)$ for $\Sigma_{\alpha} (\mathcal{A})$. In this article, using some simple combinatorial arguments, we establish optimal lower bounds for the size of $\Sigma_{\alpha} (A)$ and $\Sigma_{\alpha} (\mathcal{A})$. As special cases, we also obtain some already known results in this study.


Introduction
Let A be a set of k integers. The sum of all elements of a subset of A is called a subset sum of A. So, the subset sum of the empty set is 0. For a nonnegative integer α ≤ k, let That is, Σ α (A) is the set of subset sums corresponding to the subsets of A that are of the size at least α and Σ α (A) is the set of subset sums corresponding to the subsets of A that are of the size at most k − α. So, Σ α (A) = a∈A a − Σ α (A). Therefore |Σ α (A)| = |Σ α (A)|. Now, we extend the above definitions for sequences of integers. Before we go for extension, we mention some notation that are used throughout the paper.
Let A = (a 1 , . . . , a 1 r copies , a 2 , . . . , a 2 r copies , . . . , a k , . . . , a k r copies ) be a sequence of rk terms, where a 1 , a 2 , . . . , a k are distinct integers each appearing exactly r times in A. We denote this sequence by A = (a 1 , a 2 , . . . , a k ) r . If A ′ is a subsequence of A, then we write A ′ ⊂ A. By x ∈ A, we mean x is a term in A. For the number of terms in a sequence A, we use the notation |A|. For an integer x, we let x * A be the sequence which is obtained from by multiplying each term of A by x. For two nonempty sequences A, B, by A ∩ B, we mean the sequence of all those terms that are in both A and B. Furthermore, for integers a, b with b ≥ a, by [a, b] r , we mean the sequence (a, a + 1, . . . , b) r .
Let A = (a 1 , a 2 , . . . , a k ) r be a sequence of integers with rk terms. The sum of all terms of a subsequence of A is called a subsequence sum of A. For a nonnegative integer α ≤ rk, let That is, Σ α (A) is the set of subsequence sums corresponding to the subsequences of A that are of the size at least α and Σ α (A) is the set of subsequence sums corresponding to the subsequences of A that are of the size at most rk − α. Then in the same line with the subset sums, we have |Σ α (A)| = |Σ α (A)| for all 0 ≤ α ≤ rk.
The set of subset sums Σ α (A) and Σ α (A) and the set of subsequence sums Σ α (A) and Σ α (A) may also be written as unions of sumsets: For a finite set A of k integers and for positive integers h, r, the h-fold sumset hA is the collection of all sums of h not-necessarily-distinct elements of A, the h-fold restricted sumset hˆA is the collection of all sums of h distinct elements of A, and the generalized sumset h (r) A is the collection of all sums of h elements of A with at most r repetitions for each element (see [28]). Then Σ α (A) = k h=α hˆA, Σ α (A) = k−α h=0 hˆA, Σ α (A) = and Jiang and Li [27] extended Nathanson's results to Σ 1 (A) for sequences of integers A. Note that these subset and subsequence sums may also be studied in any abelian group (for earlier works, in case α = 0 and α = 1, see [6, 8, 18, 21-23, 25, 26, 35]). Recently, Balandraud [7] proved the optimal lower bound for |Σ α (A)| in the finite prime field F p , where p is a prime number. Inspired by Balandraud's work [7], in this paper we establish optimal lower bounds for |Σ α (A)| and |Σ α (A)| in the group of integers. Note that, in [13], we have already settled this problem when the set A (or sequence A) contains nonnegative or nonpositive integers. So, in this paper we consider the sets (or sequences) which may contain both positive and negative integers.
In Section 2, we prove optimal lower bounds for |Σ α (A)| for finite sets of integers A. In Section 3, we extend the results of Section 2 to sequences of integers.
The following results are used to prove the results in this paper.

Theorem 1. [34, Theorem 1.4] Let A, B be nonempty finite sets of integers. Set
This lower bound is optimal.
This lower bound is optimal.

Subset sum
In Theorem 4 and Corollary 1, we prove optimal lower bound for |Σ α (A)| under the assumptions A ∩ (−A) = ∅ and A ∩ (−A) = {0}, respectively. In Theorem 5 and Corollary 2, we prove optimal lower bound for |Σ α (A)| for arbitrary finite sets of integers A. The bounds in Theorem 5 and Corollary 2 depends on the number of positive elements and the number of negative elements in set A. In Corollary 3, we prove lower bounds for |Σ α (A)|, which holds for arbitrary finite sets of integers A and only depend on the total number of elements of A not the number of positive and negative elements of A.
This lower bound is optimal.
Proof. Let p be a prime number such that p > max 2 max Next, to verify that the lower bound in (1) This together with (1) gives Thus, the lower bound in (1) is optimal. This completes the proof of the theorem.
This lower bound is optimal.
This together with (2) gives that the lower bound in (2) is optimal.
Theorem 5. Let n and p be positive integers and A be a set of n negative and p positive integers. Let α ∈ [0, n + p] be an integer.
These lower bounds are optimal.
Hence, by Theorem 1 and Theorem 4, we have (ii) If α ≤ n and α > p, then Hence, by Theorem 1 and Theorem 4, we have (iii) If α > n and α ≤ p, then by applying the result of (ii) for (−A), we obtain (iv) If α > n and α > p, then Hence, by Theorem 4, we get It can be easily verified that all the lower bounds mentioned in the theorem are optimal for A = [−n, p] \ {0}.
Corollary 2. Let n and p be positive integers and A be a set of n negative integers, p positive integers and zero. Let α ∈ [0, n + p + 1] be an integer.
These lower bounds are optimal.
Remark 2. Nathanson [33] have already proved this corollary for α = 1. The purpose of this corollary is to prove a similar result for every α ∈ [0, k]. Note that the lower bounds in Corollary 3 are not optimal for all α ∈ [0, k], except for α = 0 and α = 1.
Remark 3. The lower bounds in Corollary 3 can also be written in the following form:

Subsequence sum
In this section, we extend the results of the previous section from sets of integers to sequences of integers. In Theorem 6, we establish optimal lower bound for |Σ α (A)| under the assumptions A ∩ (−A) = ∅ and A ∩ (−A) = (0) r . In Theorem 7 and Corollary 4, we prove optimal lower bound for |Σ α (A)| for arbitrary finite sequences of integers A. The bounds in Theorem 7 and Corollary 4 depends on the number of negative terms and the number of positive terms in sequence A. In Corollary 5, we prove lower bounds for |Σ α (A)|, which holds for arbitrary finite sequences of integers A and only depend on the total number of terms of A not the number of positive and negative terms of A.
Next, to verify that the lower bounds in (5) and (6) These two inequalities together with (5) and (6) implies that the lower bounds in (5) and (6) are optimal. This completes the proof of the theorem. (iv) If m > n and m > p, then |Σ α (A)| ≥ r n(n+1) These lower bounds are optimal.
Proof. Let A n and A p be sets that contain respectively, all distinct negative terms and all distinct positive terms of A. Then |A n | = n and |A p | = p. Let also A n = {b 1 , b 2 , . . . , b n } and A p = {c 1 , c 2 , . . . , c p }, where b n < b n−1 < · · · < b 1 < 0 < c 1 < c 2 < · · · < c p .
(ii) If m ≤ n and m > p, then Hence, by Theorem 1, Theorem 2, Theorem 4, and Theorem 6, we have (iii) If m > n and m ≤ p, then by applying the result of (ii) for (−A), we obtain (iv) If m > n and m > p, then Hence, by Theorem 6, we have Furthermore, the optimality of the lower bounds in (i)-(iv) can be verified by taking A = [−n, p] r \ (0) r . Corollary 4. Let k ≥ 2, r ≥ 1, and α ∈ [0, rk − 1] be integers. Let m ∈ [1, k] be an integer such that (m − 1)r ≤ α < mr. Let A be a sequence of rk terms which is made up of n negative integers, p positive integers and zero, each repeated exactly r times.
Proof. The lower bounds for |Σ α (A)| easily follows from Theorem 7 and the fact that Σ α (A) = Σ 0 (A ′ ) for 0 ≤ α < r and Σ α (A) = Σ α−r (A ′ ) for r ≤ α < rk, where A ′ = A \ (0) r . Furthermore, the optimality of these bounds can be verified by taking A = [−n, p] r . Along the same line with the sumsets hA, hˆA, and Σ α (A), the first author established optimal lower bounds for |HA| and |HˆA|, when A contains nonnegative or nonpositive integers (see [9]). The author also characterized the sets H and A for which the lower bounds are achieved [9]. It will be interesting to generalize such results to the sumset H (r) A.