EDB — 1TY

Exercises

1. [1TY]Note:written exam 12/1/2013.

   Given a subset \(E\) of \(ℕ\) and an integer \(n ∈ ℕ\), the expression

   \[ \frac{\mathrm{card}( E ∩ \{ 0, 1, . . . , n\} )}{n+1} \]

   indicates which fraction of the segment \(\{ 0, 1, . . . , n\} \) is contained in E. The notion of "density" in \(ℕ\) of \(E\) refers to the behavior of such fractions as n tends to infinity. Precisely, we define the upper density \( \overline d(E)\) of E and its lower density \(\underline d(E)\) as

   \[ \overline d(E) = \limsup _{n→∞} \frac{\mathrm{card}( E ∩ \{ 0, 1, … , n\} )}{n+1}\quad , \]
   \[ \underline d(E) = \liminf _{n→∞} \frac{\mathrm{card}( E ∩ \{ 0, 1, … , n\} )}{n+1}\quad . \]

   If \(\overline d(E) = \underline d(E) = d ∈ [0, 1]\), E is said to have density d. (See also [ 52 ] .)

   1. Prove that, for every \(α ∈ ℝ, α ≥ 1\), the set \(E_α = [nα] : n ∈ N\) has density \(d = 1/α\) (the symbol \([x]\) indicates the integer part of \(x ∈ R\)).

   2. Let \(E = \{ m_ 0 , m_ 1 , … , m_ k , … \} \) be an infinite set, with \(m_ 0 {\lt} m_ 1 {\lt} … {\lt} m_ k {\lt} …\). Prove that \(\overline d(E) = \limsup _{ k→∞} \frac{k}{m_ k}\) and \(\underline d(E) = \liminf _{ k→∞} \frac{k}{m_ k}\).

   3. Find a set E with \(\overline d(E) = \overline d(ℕ ⧵ E) = 1\).