On the cardinality of subsequence sums. II (Q6639456)

Let\N\[\NA = (a_1, a_2, \dots, a_k)\N\]\Nbe a sequence of integers, where \( a_1 < a_2 < \dots < a_k \), and each \( a_i \) is repeated \( r_i \) times, with \( r_i \geq 1 \). The sum of the elements of any subsequence \( B \) of \( A \) is called a \textit{subsequence sum} of \( A \), denoted by \( \sigma(B) \).\N\NFor \( 0 \leq \alpha \leq \sum_{i=1}^k r_i \), define\N\[\N\Sigma_\alpha(A) = \{\sigma(B) \mid B \text{ is a subsequence of } A \text{ of length } \geq \alpha\}.\N\]\NWe denote \( \Sigma_1(A) \) simply as \( \Sigma(A) \). Clearly, \( |\Sigma_\alpha(A)| = |\Sigma_\alpha(A)| \), and it is evident that \( 0 \in \Sigma_\alpha(A) \). When \( r_i = 1 \) for all \( 1 \leq i \leq k \), the sequence \( A \) becomes a set of integers, and in this case, we refer to the subsequence sums as \textit{subset sums}. Both subset sums and subsequence sums are fundamental objects in additive number theory. They are particularly useful in solving various combinatorial problems, such as zero-sum problems.\N\NThe primary challenge in studying the subsequence sums \( \Sigma_\alpha(A) \) is twofold:\N\begin{itemize}\N\item[1.] \textbf{Direct Problem:} To determine the optimal lower bound of \( |\Sigma_\alpha(A)| \), the size of \( \Sigma_\alpha(A) \).\N\item[2.] \textbf{Inverse Problem:} To identify the structure of the sequence \( A \) for which \( |\Sigma_\alpha(A)| \) is minimal.\N\end{itemize}\N\NIn 1995, \textit{M. B. Nathanson} [Trans. Am. Math. Soc. 347, No. 4, 1409--1418 (1995; Zbl 0835.11006)] was the first to investigate the direct and inverse problems for \( \Sigma(A) \) in the context of sets \( A \) of integers. Subsequently, in 2015, \textit{R. K. Mistri} et al. [J. Number Theory 148, 235--256 (2015; Zbl 1379.11025)] extended Nathanson's results to the case of subsequence sums \( \Sigma(A) \), considering sequences \( A \) consisting exclusively of nonnegative integers or exclusively of nonpositive integers. Later, the work by the present author and Li addressed the remaining case where \( A \) contains a mix of positive integers, negative integers, and/or zero.\N\NIn 2020, \textit{J. Bhanja} and \textit{R. K. Pandey} [Discrete Math. 343, No. 12, Article ID 112148, 11 p. (2020; Zbl 1458.11016)] explored the direct and inverse problems for \( \Sigma_\alpha(A) \) when the sequence \( A \) includes either nonnegative or nonpositive integers. Their results contributed significantly to understanding the structure and properties of subsequence sums in these specific cases.\N\NThis paper is a contribution to this area by studying both direct and inverse results.