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SUMMARY:Steven J. Miller (Williams College\, Fibonacci Association)
DTSTART:20250521T203000Z
DTEND:20250521T205500Z
DTSTAMP:20260423T041654Z
UID:CANT2025/38
DESCRIPTION:Title: <a href="https://researchseminars.org/talk/CANT2025/38/
 ">Phase transitions for binomial sets under linear forms</a>\nby Steven J.
  Miller (Williams College\, Fibonacci Association) as part of Combinatoria
 l and additive number theory (CANT 2025)\n\nLecture held in CUNY Graduate 
 Center - Science Center (4th floor).\n\nAbstract\nWe generalize results on
  sum and difference sets of a subset S of\n$\\mathbb{N}$ drawn from a bino
 mial model. Given $A \\subseteq \\{0\, 1\,\n\\dots\, N\\}$\, an integer $h
  \\geq 2$\, and a linear form $L: \\mathbb{Z}^h \\to\n\\mathbb{Z}$ $$L(x_1
 \, \\dots\, x_h)\\ :=\\ u_1x_1 + \\cdots + u_hx_h\, \\quad u_i\n\\in \\mat
 hbb{Z}_{\\neq 0} {\\rm\\ for\\ all\\ } i \\in [h]\,$$ we study the size\no
 f $$L(A)\\ =\\ \\left\\{u_1a_1 + \\cdots + u_ha_h : a_i \\in A \\right\\}$
 $ and\nits complement $L(A)^c$ when each element of $\\{0\, 1\, \\dots\, N
 \\}$ is\nindependently included in $A$ with probability $p(N)$\, identifyi
 ng two\nphase transitions. The first global one concerns the relative size
 s of\n$L(A)$ and $L(A)^c$\, with $p(N) = N^{-\\frac{h-1}{h}}$ as the thres
 hold.\nAsymptotically almost surely\, below the threshold almost all sums\
 ngenerated in $L(A)$ are distinct and almost all possible sums are in\n$L(
 A)^c$\, and above the threshold almost all possible sums are in $L(A)$.\nO
 ur asymptotic formulae substantially extends work of Hegarty and Miller\,\
 nresolving their conjecture. The second local phase transition concerns th
 e\nasymptotic behavior of the number of distinct realizations in $L(A)$ of
  a\ngiven value\, with $p(N) = N^{-\\frac{h-2}{h-1}}$ as the threshold and
 \nidentifies (in a sharp sense) when the number of such realizations obeys
  a\nPoisson limit. Our main tools are recent results on the asymptotic\nen
 umeration of partitions\, Stein's method for Poisson approximation\, and\n
 the martingale machinery of Kim-Vu.\n
LOCATION:https://researchseminars.org/talk/CANT2025/38/
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