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SUMMARY:Philippa Holdridge (Alfréd Rényi Institute\, Hungary)
DTSTART:20260717T160000Z
DTEND:20260717T162500Z
DTSTAMP:20260710T111505Z
UID:CANT2026/65
DESCRIPTION:Title: <a href="https://researchseminars.org/talk/CANT2026/65/
 ">The size of certain symmetric differences of sets of integers</a>\nby Ph
 ilippa Holdridge (Alfréd Rényi Institute\, Hungary) as part of Combinato
 rial and additive number theory seminar (CANT 2026)\n\nLecture held in Sci
 ence Center in the CUNY Graduate Center (4th floor).\n\nAbstract\nConsider
  a set $A\\subseteq \\mathbb{N}$ which is finite and nonempty. Letting $\\
 Delta$ denote the symmetric difference of sets and $k\\cdot A=\\{ka:a\\in 
 A\\}$\, it can be shown that $A\\Delta (2\\cdot A)$ always contains at lea
 st two elements. It also turns out that $A\\Delta (2\\cdot A) \\Delta (3\\
 cdot A)$ has at least three elements. Does $A\\Delta (2\\cdot A)\\Delta\\c
 dots \\Delta (n\\cdot A)$ have at least $n$ elements for all $n\\in \\math
 bb{N}$? This question was posed by Pilz in an equivalent form involving th
 e minimal distance of certain linear codes. If true\, then this lower boun
 d is best possible\, as seen by considering $A=\\{1\\}$. The lower bound i
 s also attained when $A=\\{1\,2\,\\dots\,n\\}$ and\, in fact\, for each $n
 $\, there are arbitrarily large sets $A$ such that $A\\Delta (2\\cdot A)\\
 Delta\\cdots \\Delta (n\\cdot A)$ has exactly $n$ elements.\n\nPilz proved
  the conjecture for $n\\le 6$\, and it can also be proven for $n=7$ and $8
 $. For larger $n$\, Pach and Szabó proved a lower bound of the form $n/(\
 \log n)^{\\lambda}$ for $\\lambda\\approx 0.22$. Until recently\, this was
  the strongest result known\, but in a recent work\, we have proven the co
 njecture for all sufficiently large $n$. \nMore precisely\, whenever $n\\g
 e 3^{81}$. In this talk we will outline the proof and discuss some related
  problems. Joint work with P. Pach.\n
LOCATION:https://researchseminars.org/talk/CANT2026/65/
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