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SUMMARY:Mohan (BK Birla Institute of Engineering and Technology\, India)
DTSTART:20250520T110000Z
DTEND:20250520T112500Z
DTSTAMP:20260423T010336Z
UID:CANT2025/1
DESCRIPTION:Title: <a href="https://researchseminars.org/talk/CANT2025/1/"
 >Special additive complements of a set of natural numbers</a>\nby Mohan (B
 K Birla Institute of Engineering and Technology\, India) as part of Combin
 atorial and additive number theory (CANT 2025)\n\nLecture held in CUNY Gra
 duate Center - Science Center (4th floor).\n\nAbstract\nLet $A$ be a set o
 f natural numbers.	A set $B$ of natural numbers is said to be an additive 
 complement of the set $A$ if all sufficiently large natural numbers can be
  represented as $x+y$ for some $x\\in A$ and $y\\in B$.  We shall describe
  various types of additive complements of the set $A$ such as those additi
 ve complements of $A$  that do or do not intersect  $A$\, additive complem
 ents which are the union of disjoint infinite arithmetic progressions\, an
 d additive complements having various densities etc. We estabilish that if
    $A=\\{a_i: i\\in \\mathbb{N}\\}$ is a set of natural numbers such that 
  $a_{i} < a_{i+1} $ for $i \\in \\mathbb{N}$ and $\\liminf_{n\\rightarrow 
 \\infty } (a_{n+1}/a_{n})>1$\, then there exists a set $B\\subset \\mathbb
 {N}$  such that $B\\cap A = \\varnothing$ and $B$ is a sparse additive com
 plement of the set $A$.  Besides this\,  for a given positive real number 
 $\\alpha \\leq 1$ and a finite set $A$\, we investigate a set $B$ such tha
 t $B$ can be written as a union of disjoint infinite arithmetic progressio
 ns with the natural density of $A+B$ equal to $\\alpha$.\n
LOCATION:https://researchseminars.org/talk/CANT2025/1/
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