Abstract
This chapter contains descriptions of 8 great theorems published in the Annals of Mathematics in 2003.
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Notes
- 1.
A function \(f:X \rightarrow Y\) is called convex if \(f(\alpha x + (1-\alpha )y) \le \alpha f(x) + (1-\alpha )f(y)\) for all \(x \in X\), \(y \in X\), and for all \(\alpha \in [0,1]\).
- 2.
If you are not familiar with the formulas for derivatives, you can verify this directly from definition (3.7): \( \lim _{\varepsilon \rightarrow 0}\frac{(t+\varepsilon )^3/3-t^3/3}{\varepsilon }=\frac{1}{3}\lim _{\varepsilon \rightarrow 0}\frac{t^3+3t^2\varepsilon +3t\varepsilon ^2+\varepsilon ^3-t^3}{\varepsilon } = \frac{1}{3}\lim _{\varepsilon \rightarrow 0}(3t^2+3t\varepsilon +\varepsilon ^2) = t^2 \).
- 3.
In fact, if we insist that all \(x_i\) are either 0 or 1, then we could select only \(S \subset S'\), where \(S'\) is the set of points with this property, but this problem is still similar to sphere packing.
- 4.
In general, the limit (3.16) may not exist, so the formal definition is more complicated. If \(A=(A_1, A_2, \dots )\) is a collection of closed sets that cover the set S, let \(C^d(A, S)=\sum _i D(A_i)^d\), and let \( C^d(S) = \inf _A C^d(A, S) = \inf _A \sum _i D(A_i)^d, \) where the infimum is taken over all possible collections A of closed sets that cover set S. Then the Hausdorff dimension of S is \( \dim (S):=\inf \{d\ge 0: C^d(S)=0\}\).
- 5.
A number \(\lambda \) is called diophantine if for every \(\varepsilon > 0\) there exists a \(C_\varepsilon > 0\) such that \(\left| \lambda - \frac{p}{q}\right| \ge \frac{C_\varepsilon }{q^{2+\varepsilon }}\) for every rational number \(\frac{p}{q}\), see Sect. 3.6 for a more detailed discussion.
- 6.
A triangle is called obtuse if it has an obtuse angle, that is, an angle larger than 90 degrees.
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Grechuk, B. (2019). Theorems of 2003. In: Theorems of the 21st Century. Springer, Cham. https://doi.org/10.1007/978-3-030-19096-5_3
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DOI: https://doi.org/10.1007/978-3-030-19096-5_3
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