Q17
Proof
Existence Proof
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The purpose of this question is to show that for any strictly positive integers $n$ and $k$, there exists a constant $\alpha(k,n) \in ]0,1[$ such that, if $t_1, \ldots, t_n$ are strictly positive real numbers satisfying $1 > \sum_{i=1}^n t_i > 1 - \alpha(k,n)$, then there exist non-negative integers $p_1, \ldots, p_n \geqslant 0$ and $q$ such that $$\sum_{i=1}^n p_i = q > 0, \quad \text{and for all } i = 1, \ldots, n, \quad (kq+1)t_i > kp_i.$$ We proceed by induction on $n$.
17a. Handle the case $n = 1$ by showing that the constant $\alpha(k,1) = \frac{1}{k+1}$ works.
We assume the statement is true up to rank $n-1 \geqslant 1$. In particular, $\alpha(k,n-1) > 0$ is defined for all $k \geqslant 1$. We set for $k \geqslant 1$ $$\alpha(k,n) = \frac{1}{4kN^{n-1}} \quad \text{where} \quad N = 1 + \max\left(\frac{4k}{\alpha(k,n-1)}, 2kn(n+1)\right).$$ We are given $t_1 \geqslant t_2 \geqslant \cdots \geqslant t_n > 0$, and we assume that $\sum_{i=1}^n t_i = 1 - \alpha$ with $0 < \alpha < \alpha(k,n)$.
17b. If $t_n < \alpha(k,n-1) - \alpha$, establish the statement at rank $n$.
17c. If $t_n \geqslant \alpha(k,n-1) - \alpha$, apply the result of question 16 to the $\frac{t_i}{1-\alpha}$, $i = 1, \ldots, n$. With its notation, show that $$\alpha(k,n) < \min\left(\frac{1}{n+1}, \frac{1}{2}\alpha(k,n-1)\right) \quad \text{and} \quad 1 - qk\frac{\alpha}{1-\alpha} \geqslant \frac{1}{2}.$$ Conclude by distinguishing the cases $i \geqslant 2$ and $i = 1$.