An improved lower bound for Erdős–Szekeres products
Abstract
In 1959, Erdős and Szekeres [Acad. Serbe Sci. Publ. Inst. Math. 13 (1959), pp. 29–34] posed a series of problems concerning the size of polynomials of the form P n ( z ) = ∏ j = 1 n ( 1 − z s j ) , \begin{equation*} P_n(z) = \prod _{j=1}^n (1 - z^{s_j}), \end{equation*} where s 1 , … , s n s_1, \dots , s_n are positive integers. Of particular interest is the quantity f ( n ) = inf s 1 , … , s n ≥ 1 max | z | = 1 | P n ( z ) | . \begin{equation*} f(n) = \inf _{s_1,\dots ,s_n\ge 1} \max _{|z|=1} |P_n(z)|. \end{equation*} They proved that lim n → ∞ f ( n ) 1 / n = 1 \lim _{n\to \infty } f(n)^{1/n} = 1 , and also established the classical lower bound
Fields of Study & Subjects
Limits and Structures in Graph Theory
Analytic Number Theory Research
Graph Labeling and Dimension Problems
Publication details
- Journal / source
- Proceedings of the American Mathematical Society
- Volume
- 154
- Issue
- 8
- Pages
- 3381–3388
- Publication date
- 5 March 2026
- Language
- EN