BEGIN:VCALENDAR
VERSION:2.0
PRODID:researchseminars.org
CALSCALE:GREGORIAN
X-WR-CALNAME:researchseminars.org
BEGIN:VEVENT
SUMMARY:Mel Nathanson (CUNY)
DTSTART:20260312T190000Z
DTEND:20260312T203000Z
DTSTAMP:20260427T053038Z
UID:New_York_Number_Theory_Seminar/134
DESCRIPTION:Title: <a href="https://researchseminars.org/talk/New_York_Num
 ber_Theory_Seminar/134/">Sumset intersection problems</a>\nby Mel Nathanso
 n (CUNY) as part of New York Number Theory Seminar\n\n\nAbstract\nLet $\n 
 = \\{1\,2\,3\,\\ldots\\}$ be the set of positive integers. Let $A$ be a su
 bset of  an additive abelian semigroup $S$ and let $hA$ be the $h$-fold su
 mset of $A$.   The following question is considered:\nLet $(A_q)_{q=1}^{\\
 infty}$ be a strictly decreasing sequence of sets in\n$S$ and let  $A = \\
 bigcap_{q=1}^{\\infty} A_q$.  \nDescribe the set  $\\mathcal{H}(A_q)$  of 
 positive integers such that\n\\[\nhA = \\bigcap_{q=1}^{\\infty} hA_q.\n\\]
 \nA  sample result:  If $A_q$ is a set of positive integers for all $q$\, 
 then $\\mathcal{H}(A_q)=\n$.\n\nHere are some nice open problems.  \n\n1. 
 For a given set $X$\, does there exist a  strictly decreasing sequence\n$(
 A_q)_{q=1}^{\\infty} $ of sets of integers such that $\\mathcal{H}(A_q)= X
 $.  \n\n2. For   given sets $A$ and  $X$\, does there exist a  strictly de
 creasing sequence\n$(A_q)_{q=1}^{\\infty} $ of sets of integers such that 
 $A = \\bigcap_{q=1}^{\\infty} A_q$\nand $\\mathcal{H}(A_q)= X$.  \n\n3. Do
 es there exist a set $Y$ of positive integers such that $\\mathcal{H}(A_q)
  \\neq Y$\nfor every strictly decreasing sequence $(A_q)_{q=1}^{\\infty}$ 
 of sets of integers?\n\n4. For a given set $A$\, let $\\mathcal{H}^*(A)$ b
 e the set of all sets $X$  such that $\\mathcal{H}(A_q) = X$ for some  str
 ictly decreasing sequence $(A_q)_{q=1}^{\\infty}$ with $A = \\bigcap_{q=1}
 ^{\\infty} A_q$.\n\n5. Compute $\\mathcal{H}^*(A)$ for $A = \\{0\\}$.\n
LOCATION:https://researchseminars.org/talk/New_York_Number_Theory_Seminar/
 134/
END:VEVENT
END:VCALENDAR