For a set \( A \) of positive integers \( a_1 < a_2 < \cdots \), let \( \underline{d}(A) \) and \( \overline{d}(A) \) denote its lower and upper asymptotic densities. The gap density is defined as

\[
\lambda(A) = \limsup_{n \to \infty} \frac{a_{n+1}}{a_n}.
\]

The paper investigates the class \( \mathcal{G}(\alpha, \beta, \gamma) \) of all sets \( A \) with \( \underline{d}(A) = \alpha \), \( \overline{d}(A) = \beta \) and \( \lambda(A) = \gamma \) for given \( \alpha, \beta, \gamma \) with \( 0 \leq \alpha \leq \beta \leq 1 \leq \gamma \) and \( \alpha\gamma \leq \beta \). Using the classical dyadic mapping \( \varrho(A) = \sum_{n=1}^{\infty} \dfrac{\chi_A(n)}{2^n} \), where \( \chi_A \) is the characteristic function of \( A \), the main result of the paper states that the \( \varrho \)-image set \( \varrho\mathcal{G}(\alpha, \beta, \gamma) \) has the Hausdorff dimension

\[
\dim \varrho\mathcal{G}(\alpha, \beta, \gamma) = \min\left\{ \delta(\alpha), \delta(\beta), \frac{1}{\gamma} \max_{\sigma \in [\alpha\gamma, \beta]} \delta(\sigma) \right\},
\]

where \( \delta \) is the entropy function

\[
\delta(x) = -x\log_2 x - (1-x)\log_2(1-x).
\]