Zeros of linear recurrence sequences (Q2714159)

A sequence \(U=\{u_n\}_{n\in \mathbb Z}\) of complex numbers is called a linear recurrence sequence if it satisfies a relation \(u_n=c_1u_{n-1}+\dots+c_t u_{n-t}\) with \(c_i\in \mathbb C\) and \(c_t\neq 0\). There is only one such relation for which \(t\) is minimal. Given this recurrence relation of minimal length, we call \(t\) the order of the sequence \(U\), and \(P_U(z)=z^t-c_1z^{t-1}-\dots-c_t\) the companion polynomial of \(U\). Writing \(P_U(z)=(z-\alpha_1)^{t_1}\dots (z-\alpha_k)^{t_k}\) we have \(u_n=P_1(n)\alpha^n_1+\dots+P_k(n)\alpha^n_k\) for \(n\in \mathbb Z\), where \(P_i(z)\) is a polynomial of degree \(<t_i\) for \(i=1,\dots,k\). Let \(Z(U)\) be the set of \(n\) with \(u_n=0\). By the Skolem-Mahler-Lech theorem, \(Z(U)\) is the union of finitely many single numbers and finitely many arithmetic progressions. Consequenly, if the sequence \(U\) is nondegenerate, that is if none of the quotients \(\alpha_i/\alpha_j\) with \(j\neq j\) is a root of unity, then \(Z(U)\) is finite. NEWLINENEWLINENEWLINEIt was a long standing conjecture that if \(U\) is non-degenerate then the cardinality of \(Z(U)\) is bounded above by a quantity depending on the order \(t\) only. This was shown in several special cases by Beukers, Beukers and Tijdeman, and Schlickewei. Then Schlickewei, the author and the reviewer proved the conjecture in the special case that all polynomials \(P_i(n)\) are constant. Finally, the author [Acta Math. 182, 243-282 (1999; Zbl 0974.11013)] proved the conjeceture in full generality. More precisely he proved that if \(U\) is non-degenerate then \(Z(U)\) has cardinality \(\leq \exp \exp\exp(3t \log t)\). His proof uses a suitable quantitative version of the Subspace Theorem by Schlickewei and the reviewer but in addition to this required a lot of ingenuity. NEWLINENEWLINENEWLINEIn the present paper, the author considers also degenerate sequences. By a refinement of his proof of his last mentioned result, he proves the following quantitative version of the general theorem of Skolem-Mahler-Lech: if \(U\) is an arbitrary linear recurrence sequence of order \(t\), then \(Z(U)\) is the union of not more than \(c_1\) single numbers and \(c_2\) arithmetic progressions, where \(c_1+c_2\leq\exp \exp \exp(20t)\). In the particular case that \(U\) is non-degenerate (which implies \(c_2=0\)) this gives a slight improvement upon the author's previous result. It should be noted that whereas the original qualitative proof of the Skolem-Mahler-Lech theorem is based on \(p\)-adic analysis, the author's quantitative proof has no involvement of \(p\)-adic analysis whatsoever.