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.

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