Share of fortunate people in some pie splitting setting

Question

(This question is a follow-up on an older one.)

A huge pie is divided among $N$ guests. The first guest gets $\frac{1}{N}$ of the pie. Guest number $k$ guest gets $\frac{k}{N}$ of what's left, for all $1\leq k\leq N$. (In particular, the last guest gets all of what is left.)

A guest is said to be fortunate if his share of pie is strictly greater than the average share (which is $1/N$ of the original pie). Let $f(N)$ denote the number of fortunate guests out of the total of $N$ guests. What is the value of $$\lim\sup_{N\to\infty}\frac{f(N)}{N}$$ ?

Answer 1

The $p$-th guest receives a share of $$s_{p,N}=p N^{-p-1} (N-p+1) \frac{\Gamma(N+1)}{\Gamma(N-p+2)}$$ For large $N$ I substitute the asymptotic expansion of the Gamma function, $$\Gamma(z)\approx e^{-z}z^z(2\pi/z)^{1/2},$$ to arrive at the approximation $$s_{p,N}\approx p(N-p+1) e^{1-p} \sqrt{\frac{1}{N+1}}\sqrt{\frac{1}{N-p+2}} (N+1)^{N+1} N^{-p-1} (N-p+2)^{-N+p-1}$$ which agrees very well with the exact value (compare the dots with the continuous curve in the plot below, for $N=1000$)

Now the desired number $f(N)$ of fortunate guests should follow by solving $s_{p,N}=1/N$ for $p\equiv f$. As a check, for $N=1000$ the exact integer result is $f=94$, while the large-$N$ asymptotics gives $f=94.334$.

I would have guessed $f\approx \sqrt{N}$, but numerically I find a slightly more rapid increase, the plot below is a comparison with $f=\sqrt{2N}\log\log N$. In any case, the large-$N$ limit of $f/N$ seems to be zero.

Gold curve: Number $f$ of fortunate guests versus $N$, calculated form the asymptotics of $s_{p,N}$. The blue curve is $f=\sqrt{2N}\log\log N$.