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SUMMARY:Mikhail Gabdullin (Steklov Mathematical Institute\, Moscow\, Russi
 a)
DTSTART:20220527T143000Z
DTEND:20220527T145500Z
DTSTAMP:20260423T011437Z
UID:CANT2022/57
DESCRIPTION:Title: <a href="https://researchseminars.org/talk/CANT2022/57/
 ">A conjecture of Cilleruelo and Cordoba and divisors in a short interval<
 /a>\nby Mikhail Gabdullin (Steklov Mathematical Institute\, Moscow\, Russi
 a) as part of Combinatorial and additive number theory (CANT 2022)\n\n\nAb
 stract\nLet $E(A)=\\#\\{(a_1\,a_2\,a_3\,a_4)\\in A^4: a_1+a_2=a_3+a_4\\}$ 
 denote the additive energy of a set $A\\subset \n$\, and let $\\mathbb{T}=
 \\R/\\Z$ and $\\|f\\|_4=\\left(\\int_{\\mathbb{T}}|f(t)|^4dt\\right)^{1/4}
 $. It is well-known that \n$$\nE(\\{n^2: n\\leq N\\})=\\left\\|\\sum_{n\\l
 eq N}e^{2\\pi in^2x}\\right\\|_4^4 \\asymp N^2\\log N\,\n$$\nwhile we triv
 ially have $E(A)\\geq |A|^2$. In 1992\, J. Cilleruelo and A. Cordoba prove
 d that $E(\\{n^2: N\\leq n\\leq N+N^{\\gamma}\\})\\asymp N^{2\\gamma}$ for
  any $\\gamma\\in (0\,1)$\, and conjectured a much more general bound (aga
 in\, for any $\\gamma\\in(0\,1)$)\n$$\n\\left\\|\\sum_{N\\leq n\\leq N+N^{
 \\gamma}}a_ne^{2\\pi in^2x}\\right\\|_4\\leq C(\\gamma)\\left(\\sum_{N\\le
 q n\\leq N+N^{\\gamma}}|a_n|^2\\right)^{1/2}.\n$$\nWhile this bound is eas
 y to prove for $\\gamma\\leq 1/2$\, it seems to be open for any $\\gamma>1
 /2$. We prove this for all $\\gamma<\\frac{\\sqrt5-1}{2}=0.618...$ and pre
 sent a connection between this problem and a conjecture of I. Ruzsa: for a
 ny $\\epsilon>0$ there exists $C(\\epsilon)>0$ such that any positive inte
 ger $N$ has at most $C(\\epsilon)$ divisors in the interval $[N^{1/2}\, N^
 {1/2}+N^{1/2-\\epsilon}]$.\n
LOCATION:https://researchseminars.org/talk/CANT2022/57/
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