Intervals without primes near an iterated linear recurrence sequence

Abstract: Let $M$ be a fixed positive integer. Let $(R_{j}(n)){n\ge 1}$ be a linear recurrence sequence for every $j=0,1,\ldots, M$, and we set $f(n)=(R_0\circ \cdots \circ R_M)(n)$, where $(S\circ T)(n)= S(T(n))$. In this paper, we obtain sufficient conditions on $(R{0}(n)){n\ge 1},\ldots, (R{M}(n)){n\ge 1}$ so that the intervals $(|f(n)|-c\log n, |f(n)|+c\log n)$ do not contain any prime numbers for infinitely many integers $n\ge 1$, where $c$ is an explicit positive constant depending only on the orders of $R_0,\ldots, R_M$. As a corollary, we show that if for each $j=1,2,\ldots, M$, the sequence $(R_j(n)){n\ge 1}$ is positive, strictly increasing, and the constant term of its characteristic polynomial is $\pm 1$, then for every Pisot or Salem number $\alpha$, the numbers $\lfloor \alpha^{(R_1\circ \cdots \circ R_M)(n)} \rfloor $ are composite for infinitely many integers $n\ge 1$.