We study the existence of various sign and value patterns in sequences defined by multiplicative functions or related objects. For any set $A$ whose indicator function is ‘approximately multiplicative’ and uniformly distributed on short intervals in a suitable sense, we show that the density of the pattern $n+1\in A$, $n+2\in A$, $n+3\in A$ is positive as long as $A$ has density greater than $\frac{1}{3}$. Using an inverse theorem for sumsets and some tools from ergodic theory, we also provide a theorem that deals with the critical case of $A$ having density exactly $\frac{1}{3}$, below which one would need nontrivial information on the local distribution of $A$ in Bohr sets to proceed. We apply our results first to answer in a stronger form a question of Erdős and Pomerance on the relative orderings of the largest prime factors $P^{+}(n)$, $P^{+}(n+1),P^{+}(n+2)$ of three consecutive integers. Second, we show that the tuple $(\unicode[STIX]{x1D714}(n+1),\unicode[STIX]{x1D714}(n+2),\unicode[STIX]{x1D714}(n+3))~(\text{mod}~3)$ takes all the $27$ possible patterns in $(\mathbb{Z}/3\mathbb{Z})^{3}$ with positive lower density, with $\unicode[STIX]{x1D714}(n)$ being the number of distinct prime divisors. We also prove a theorem concerning longer patterns $n+i\in A_{i}$, $i=1,\ldots ,k$ in approximately multiplicative sets $A_{i}$ having large enough densities, generalizing some results of Hildebrand on his ‘stable sets conjecture’. Finally, we consider the sign patterns of the Liouville function $\unicode[STIX]{x1D706}$ and show that there are at least $24$ patterns of length $5$ that occur with positive upper density. In all the proofs, we make extensive use of recent ideas concerning correlations of multiplicative functions.

Let $q\in (1,2)$. A $q$-expansion of a number $x$ in $[0,1/(q-1)]$ is a sequence $({\it\delta}_{i})_{i=1}^{\infty }\in \{0,1\}^{\mathbb{N}}$ satisfying

$$\begin{eqnarray}x=\mathop{\sum }_{i=1}^{\infty }\frac{{\it\delta}_{i}}{q^{i}}.\end{eqnarray}$$

${\mathcal{B}}_{\aleph _{0}}$

$q$

$x$

$q$

${\mathcal{B}}_{1,\aleph _{0}}$

$q$

$1$

$q$

$1=\sum _{i=1}^{\infty }q^{-n_{i}}$

$\min {\mathcal{B}}_{\aleph _{0}}=\min {\mathcal{B}}_{1,\aleph _{0}}=(1+\sqrt{5})/2$

${\mathcal{B}}_{\aleph _{0}}\cap ((1+\sqrt{5})/2,q_{1}]=\{q_{1}\}$

$q_{1}\,({\approx}1.64541)$

$x^{6}-x^{4}-x^{3}-2x^{2}-x-1=0$

${\mathcal{B}}_{1,\aleph _{0}}$

$q_{3}\,({\approx}1.68042)$

$x^{5}-x^{4}-x^{3}-x+1=0$

${\mathcal{B}}_{\aleph _{0}}\cap (q_{1},q_{3}]=\{q_{2},q_{3}\}$

$q_{2}\,({\approx}1.65462)$

$x^{6}-2x^{4}-x^{3}-1=0$

Fix an integer $N\ge 2$. For a positive integer $n\in\Bbb N$, let $n=d_{0}(n)+d_{1}(n)N+d_{2}(n)N^{2}+\ldots+d_{\gamma(n)}(n)N^{\gamma(n)}$ where $d_{i}(n)\in\{0,1,2,\ldots,N-1\}$ and $d_{\gamma(n)}(n)\not=0$ denote the $N$-ary expansion of $n$. For a probability vector $\bold p=(p_{0},\ldots,p_{N-1})$ and $r>0$, the $r$ approximative discrete Besicovitch–Eggleston set $B_{r}(\bold p)$ is defined by $$B_{r}(\bold p) = \bigg\{n\in\Bbb N\,{|}\,\bigg\|\,\frac{|\{0\le k\le\gamma(n)\,{|}\,d_{k}(n)=i\}|}{\gamma(n)\,{+}\,1} - p_{i} \bigg| \le r \,\,\text{for all $i$} \bigg\}\,,$$ that is, $B_{r}(\bold p)$ is the set of positive integers $n$ such that the frequency of the digit $i$ in the $N$-ary expansion of $n$ differs from $p_{i}$ by less than $r$ for all $i\in\{0,1,2,\ldots,N-1\}$. Three natural fractional dimensions of subsets $E$ of ${\mathbb N}$ are defined, namely, the lower fractional dimension $\underline\dim(E)$, the upper fractional dimension $\overline\dim(E)$ and the exponent of convergence $\delta(E)$, and the dimensions of various subsets of ${\mathbb N}$ defined in terms of the frequencies of the digits in the $N$-ary expansion of the positive integers are studied. In particular, the dimensions of $B_{r}(\bold p)$ are computed (in the limit as $r\searrow 0$). Let ${\bold p}=(p_{0},\ldots,p_{N-1})$ be a probability vector. Then $$\lim_{r\searrow 0}\,\,\underline{\dim}(B_{r}({\bold p})) = \lim_{r\searrow 0}\,\,\overline{\dim}(B_{r}({\bold p})) = \lim_{r\searrow 0}\,\,\delta(B_{r}({\bold p})) = -\frac{\sum_{i}p_{i}\log p_{i}}{\log N}\,.$$ This result provides a natural discrete analogue of a classical result due to Besicovitch and Eggleston on the Hausdorff dimension of certain sets of non-normal numbers.

Several applications to the theory of normal numbers are given.