An improved lower bound for Erdős--Szekeres products
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866914050272657408 |
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| author | Tang, Quanyu |
| author_facet | Tang, Quanyu |
| contents | In 1959, Erdős and Szekeres posed a series of problems concerning the size of polynomials of the form $$ P_n(z) = \prod_{j=1}^n (1 - z^{s_j}), $$ where $s_1, \dots, s_n$ are positive integers. Of particular interest is the quantity $$ f(n) = \inf_{s_1,\dots,s_n\ge 1} \max_{|z|=1} |P_n(z)|. $$They proved that $\lim_{n\to\infty} f(n)^{1/n} = 1$, and also established the classical lower bound $f(n) \ge \sqrt{2n}$. However, despite extensive effort over more than six decades, no stronger general lower bound had been established.
In this paper, we obtain the new bound $$ f(n) \ge 2\sqrt{n}. $$This gives the first improvement of the classical lower bound for the Erdős--Szekeres problem in the general case since 1959. In particular, our result confirms a remark of Billsborough et al., who observed that if the original Erdős--Szekeres proof could be fixed, the O'Hara--Rodriguez bound would yield exactly this inequality. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2509_14182 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | An improved lower bound for Erdős--Szekeres products Tang, Quanyu Number Theory Classical Analysis and ODEs Combinatorics Primary 30C10, 26D05 In 1959, Erdős and Szekeres posed a series of problems concerning the size of polynomials of the form $$ P_n(z) = \prod_{j=1}^n (1 - z^{s_j}), $$ where $s_1, \dots, s_n$ are positive integers. Of particular interest is the quantity $$ f(n) = \inf_{s_1,\dots,s_n\ge 1} \max_{|z|=1} |P_n(z)|. $$They proved that $\lim_{n\to\infty} f(n)^{1/n} = 1$, and also established the classical lower bound $f(n) \ge \sqrt{2n}$. However, despite extensive effort over more than six decades, no stronger general lower bound had been established. In this paper, we obtain the new bound $$ f(n) \ge 2\sqrt{n}. $$This gives the first improvement of the classical lower bound for the Erdős--Szekeres problem in the general case since 1959. In particular, our result confirms a remark of Billsborough et al., who observed that if the original Erdős--Szekeres proof could be fixed, the O'Hara--Rodriguez bound would yield exactly this inequality. |
| title | An improved lower bound for Erdős--Szekeres products |
| topic | Number Theory Classical Analysis and ODEs Combinatorics Primary 30C10, 26D05 |
| url | https://arxiv.org/abs/2509.14182 |