Problem 104 — Visual explainer (undergrad)
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Problem statement
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Given $n$ points in $\mathbb{R}^2$ the number of distinct unit circles containing at least three points is $o(n^2)$.

Picture
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Unit circle picture (radius 1):

       o
    .-'''-.
  o(  ( )  )o    points on the same unit circle
    '-...-'
       o

Conjecture / Task: what to look at
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- Restate the problem as: given the constraint, what asymptotic bound or
  structure must follow?
- Use the comments as benchmarks (best known bounds / constructions).

Context
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Area hint: incidence geometry / combinatorial geometry.

Notation (if it appears above)
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- `o(g(n))` means "grows strictly smaller than g(n)" (ratio → 0 as n→∞).

Benchmarks / known results (from comments)
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- In Erd\H{o}s proved that $\gg n$ many circles is possible, and that there
  cannot be more than $O(n^2)$ many circles.
- The argument is very simple: every pair of points determines at most $2$
  unit circles, and the claimed bound follows from double counting.
- Erd\H{o}s claims in a number of places this produces the upper bound
  $n(n-1)$, but Harborth and Mengerson note that in fact this delivers an
  upper bound of $\frac{n(n-1)}{3}$.
- Elekes has a simple construction of a set with $\gg n^{3/2}$ such circles.