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SUMMARY:Jinhui Fang (Nanjing Normal University\, China)
DTSTART:20250522T130000Z
DTEND:20250522T132500Z
DTSTAMP:20260423T010132Z
UID:CANT2025/5
DESCRIPTION:Title: <a href="https://researchseminars.org/talk/CANT2025/5/"
 >On  bounded unique representation bases</a>\nby Jinhui Fang (Nanjing Norm
 al University\, China) as part of Combinatorial and additive number theory
  (CANT 2025)\n\nLecture held in CUNY Graduate Center - Science Center (4th
  floor).\n\nAbstract\nFor a nonempty set $A$ of integers and an integer $n
 $\,  let $r_{A}(n)$ be the number of representations of $n=a+a'$ with $a\
 \le a'$ and $a\, a'\\in A$\, and let $d_{A}(n)$ be the number of represent
 ations of $n=a-a'$ with $a\, a'\\in A$. In 1941\, Erd\\H{o}s and Tur\\'{a}
 n posed the profound conjecture: If $A$ is a set of positive integers such
  that $r_A(n)\\ge 1$ for all sufficiently large $n$\, then $r_A(n)$ is unb
 ounded. In 2004\, Ne\\v{s}et\\v{r}il and Serra introduced the notion of bo
 unded sets and confirmed the Erd\\H{o}s-Tur\\'{a}n conjecture for all boun
 ded bases. In 2003\, Nathanson considered the existence of the set $A$ wit
 h logarithmic growth such that $r_A(n)=1$ for all integers $n$. Recently\,
  we prove that\, for any positive function $l(x)$ with $l(x)\\rightarrow 0
 $ as $x\\rightarrow \\infty$\,  there is a bounded set $A$ of integers su
 ch that $r_A(n)=1$ for all integers $n$ and $d_A(n)=1$ for all positive in
 tegers $n$\, and $A(-x\,x)\\ge l(x)\\log x$ for all sufficiently large $x$
 \, where $A(-x\,x)$ is the number of elements $a\\in A$ with $-x\\le a\\le
  x$.  \nThis is joint work with Prof. Yong-Gao Chen.\n
LOCATION:https://researchseminars.org/talk/CANT2025/5/
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