On the linear combination of \(q\)-additive functions at prime places (Q955194)

Let \(q\geq 2\) be a fixed integer, \(\mathcal{A}=\{0,1,\dots q-1\}\). For each \(n\in \mathbb{N}_0=\{0,1,\dots\}\) the expansion \[ n=\sum_{j=0}^\infty \varepsilon_j(n) q^j \] holds with some \(\varepsilon_j(n)\in \mathcal{A}\). A function \(f:\mathbb{N}_0\rightarrow \mathbb{R}\) is called \(q\)--additive if \(f(0)=0\) and \[ f(n)=\sum_{j=0}^\infty f\left(\varepsilon_j(n) q^j\right) \] for each \(n\in \mathbb{N}\). Let \(1\leq a_1<a_2<\dots <a_k<q\) be mutually coprime integers, each of which is coprime to \(q\) as well, and let \(f_1,f_2, \dots , f_k\) be \(q\)--additive functions. For collections \(a_1,a_2,\dots, a_k\) and \(f_1,f_2,\dots, f_k\) on can construct the following linear combination \[ l(n)=f_1(a_1n)+f_2(a_2n)+\dots +f_k(a_kn). \] The main result of the paper is deal to the tightness of linear combinations \(l\) on the set of prime numbers. We note only that \(l:\mathbb{N}_0\rightarrow\mathbb{R}\) is tight on the set \(\mathcal{P}\) if for suitable sequence of real numbers \(A_N\) \[ \limsup_{N\rightarrow\infty}\frac1{\pi\left(q^N\right)}\,\# \{ p<q^N:\,|l(p)-A_N|>K\}\mathop{\rightarrow}\limits_{K\rightarrow\infty} 0, \] where, as usual, \(\pi(x)\) denote the number of primes up to \(x\).