Proposition 8.14 -- the riffle shuffle needs $\log_2 n$ shuffles

The riffle shuffle (Gilbert–Shannon–Reeds) of a deck of $n$ cards cuts the deck into two packets and interleaves them; formally it is the time reversal of the inverse riffle, in which each card independently receives a uniform bit and the cards labeled $0$ move to the top preserving relative order. Its stationary distribution is uniform. For a tolerance $\varepsilon$, the mixing time $t_{\mathrm{mix}}(\varepsilon)$ is the first $t$ with $\max_x\|P^t(x,\cdot)-\mathrm{unif}\|_{TV}\le\varepsilon$, where $\|\mu-\nu\|_{TV}=\max_A|\mu(A)-\nu(A)|$.

The theorem (Proposition 8.14 of Levin–Peres–Wilmer) asserts: for any fixed tolerances $0<\varepsilon<1$ and margin $0<\delta<1$ there is an $N$ such that for all $n\ge N$,

So the upper bound $2\log_2(4n/3)+1$ of the companion theorem is sharp up to the constant factor $2$: no fixed number of riffle shuffles suffices for all deck sizes, and $\log_2 n$ is the true order. The obstruction is counting: $t$ shuffles produce at most $2^{nt}$ equally likely bit-histories, too few to spread mass over $n!$ orderings until $t\gtrsim\log_2 n$.