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SUMMARY:Sergei Konyagin (Steklov Institute\, Moscow\, Russia)
DTSTART:20230524T143000Z
DTEND:20230524T145500Z
DTSTAMP:20260423T010422Z
UID:CANT2023/24
DESCRIPTION:Title: <a href="https://researchseminars.org/talk/CANT2023/24/
 ">Large gaps between sums of two squarefull numbers</a>\nby Sergei Konyagi
 n (Steklov Institute\, Moscow\, Russia) as part of Combinatorial and addit
 ive number theory (CANT 2023)\n\nLecture held in Room 4102 in CUNY Grad Ce
 nter.\n\nAbstract\nA positive integer $n$ is called squarefull or powerful
  if in its factorization\n$n = p_1^{\\alpha_1}\\dots p_r^{\\alpha_r}$ we h
 ave $\\alpha_i\\ge2$ for all $i$.\nWe consider that $0$ is also a squarefu
 ll number.\nThus\, a number is squarefull if and only if it can be represe
 nted as\n$n = a^2b^3$ for some $a\,b\\in{\\Bbb Z}_+$.\n\nLet $W$ be the se
 t of all sums of two squarefull numbers. Blomer (2005) proved that\n$$ W(x
 ):= |W\\cap[1\,x]|  = \\frac x{(\\log x)^{\\alpha+o(1)}}\\quad(x\\to\\inft
 y)\,$$\nwhere $\\alpha = 1 - 2^{-1/3} = 0.20\\dots$.\n\nAs suggested by Sh
 parlinski\, we study large gaps between elements of $W$. \nNamely\, for $x
  > 1$ define $M(x)$ as the length of the largest subinterval of $[1\,x]$\n
 without elements of $W$. Blomer's result implies that $M(x) \\ge (\\log x)
 ^{\\alpha+o(1)}$\nas $x\\rightarrow \\infty$ since the largest gap is at l
 east as large as the average gap. We improve\nthis estimate.\n\n Theorem: 
 For $x\\ge3$ we have $M(x)\\ge c(\\log x)/(\\log\\log x)^2$ where $c>0$\ni
 s an absolute constant.\n\nJoint work with Alexander Kalmynin.\n
LOCATION:https://researchseminars.org/talk/CANT2023/24/
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