# Asymptotic densities of certain sets of numbers.

This note will establish formulae for the asymptotic densities in the natural numbers of two sets of numbers.  These sets are $S_{1}(\, q, r, s, t )\, = \{ (\, qi + r )\, q^{sj + t} : i, j \in \mathbb{N} \} \quad$

and

$S_{2}(\, q, r, s, t )\, = \{ (\, qi + r )\, q^{sj + t} : i, j \in \mathbb{N}, j > 0 \}$

for integers $\quad q, r, s, t$.

Theorem 1. Let $\quad q, r, s, t \in \mathbb{Z} \quad$ with $\quad q, r, s > 0 \quad$ and $\quad 0 < r < q \quad$. Then the asymptotic density of the set $S_{1} \quad$ is $\quad (\, q^{t + 1 - s} (\, q^{s} - 1 )\, )\, ^{-1} \quad$. In addition the asymptotic density of the set $\, S_{2} \,$ is $\, (\, q^{t + 1} (\, q^{s} - 1 )\, )\, ^{-1}$.

Proof.

We firstly look at $S_{1}$. We have, for fixed $j \geq 0$,

$\#\{ i \geq 0: (\, qi + r)\, q^{sj + t} \leq N \} = \frac {N}{q^{qj+t+1}} - \frac {r}{q} - E(\, j, N, q, r, s, t )\,$

where $0 \leq E( \, j, N, q, r, s, t ) < 1$ is an error term introduced by not rounding down to the nearest integer. So, letting

$U(N, s, t) =: \quad \lfloor \frac {\log_q (\, N )\, - t - 1}{s} \rfloor \quad \,$,

we have

$\#\{n < N: n = (\, qi + r)\, q^{sj + t} \, for some \, i, j \in \mathbb{N} \}$

$= \sum_{j \geq 0} (\, \frac {N}{q^{sj + t + 1}} - \frac {r}{q} - E(\, j, N, q, r, s, t )\, )\, = \sum_{j = 0}^{U} (\, \frac {N}{q^{sj+t + 1}} - \frac {r}{q} - E(\, j, N, q, r, s, t )\, )\,$

$= \frac {N}{q^{t + 1} } \sum_{j = 0}^{U} (\, \frac {1}{q^{s}} )\,^{j} - E^{'}(\, N, q, r, s, t )\,$

where the new error term $E^{'} (\, N, q, r, s, t )\,$ satisfies $0 < E^{'} (\, N, q, r, s, t )\, < 2(\, U + 1 )\,$.

Then

$\#\{n < N: n = (\, qi + r)\, q^{sj + t} \, for some \, i, j \in \mathbb{N} \}$

$= (\, \frac {N}{q^{t+1}} )\, (\, 1 - (\, \frac {1}{q^{s}} )\, ^{U + 1} )\, (\, 1 - \frac {1}{q^{s} } )\, ^{-1} - E^{'}$

Since $\lim_{N \to \infty} \frac {E^{'}(\, N, q, r, s, t )\, }{N} = 0$ and $\lim_{N \to \infty} \frac {1}{N} (\, \frac {1}{q^{s} } )\, ^{U + 1} = 0$

$\lim_{N \to \infty} \frac {1}{N} \#\{n < N: n = (\, qi + r)\, q^{sj + t} \, for some \, i, j \in \mathbb{N} \}$

$= (\, q^{t + 1 - s} (\, q^{s} - 1 )\,)\, ^{-1}$.

The proof for $S_{2}$ is very similar. With $E, E^{'}$ and $U$ defined as above we have for fixed $j \geq 1$,

$\#\{ i \geq 0: (\, qi + r)\, q^{sj + t} \leq N \} = \frac {N}{q^{qj+t+1}} - \frac {r}{q} - E(\, j, N, q, r, s, t )\,$

and

$\#\{n < N: n = (\, qi + r)\, q^{sj + t} \, for some \, i, j \in \mathbb{N} \, with \, j \geq 1 \}$

$= \sum_{j \geq 1} (\, \frac {N}{q^{sj + t + 1}} - \frac {r}{q} - E(\, j, N, q, r, s, t )\, )\, = \sum_{j = 1}^{U} (\, \frac {N}{q^{sj+t + 1}} - \frac {r}{q} - E(\, j, N, q, r, s, t )\, )\,$

$= \frac {N}{q^{t + 1 + s} } \sum_{j = 0}^{U - 1} (\, \frac {1}{q^{s}} )\,^{j} - E^{'}(\, N, q, r, s, t )\,$

$= (\, \frac {N}{q^{t+1+s}} )\, (\, 1 - (\, \frac {1}{q^{s}} )\, ^{U} )\, (\, 1 - \frac {1}{q^{s} } )\, ^{-1} - E^{'}$

and

$\lim_{N \to \infty} \frac {1}{N} \#\{n < N: n = (\, qi + r)\, q^{sj + t} \, for some \, i, j \in \mathbb{N} \, with \, j \geq 1 \}$

$= (\, q^{t + 1 } (\, q^{s} - 1 )\,)\, ^{-1}$.

$\Box$.