학술논문
Upper and lower bounds on the size of $B_k[g]$ sets.
Document Type
Journal
Author
Johnston, Griffin (1-VLNV-MS) AMS Author Profile; Tait, Michael (1-VLNV-MS) AMS Author Profile; Timmons, Craig (1-CASSC) AMS Author Profile
Source
Subject
11 Number theory -- 11P Additive number theory; partitions
11P70Inverse problems of additive number theory, including sumsets
11P70
Language
English
Abstract
Summary: ``A subset $A$ of the integers is a $B_k[g]$ set if the number of $k$-element multisets from $A$ that sum to any fixed integer is at most $g$. Let $F_{k,g}(n)$ denote the maximum size of a $B_k[g]$ set in $\{1,\dots,n\}$. In this paper we improve the best-known upper bounds on $F_{k,g}(n)$ for $g > 1$ and $k$ large. When $g = 1$ we match the best upper bound of Green with an improved error term. Additionally, we give a lower bound on $F_{k,g}(n)$ that matches a construction of Lindström while removing one of the hypotheses.''