An Improvement of Konstantoulas' Density Constant
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866913172247543808 |
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| author | Li, Huixi Zhang, Zihan |
| author_facet | Li, Huixi Zhang, Zihan |
| contents | Let $A\subset N$, and define its ordered representation function $r(n)=\#\{(a,b)\in A\times A:a+b=n\}.$ The Erdos--Turan conjecture asserts that, if $r(n) > 0$ for all sufficiently large $n$, then $r(n)$ is unbounded. Konstantoulas proved a density-theoretic version: if the upper density of $E=N\setminus(A+A)$ is less than $1/10$, then $\limsup_{n\to\infty} r(n)> 5$. In this paper, we improve Konstantoulas' constant to $7/32$. We also prove that $D(E)< 1/2$ implies $\limsup_{n\to\infty} r(n) > 3$, and give a conditional criterion forcing $\limsup_{n\to\infty} r(n)>7$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2605_30922 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | An Improvement of Konstantoulas' Density Constant Li, Huixi Zhang, Zihan Number Theory Let $A\subset N$, and define its ordered representation function $r(n)=\#\{(a,b)\in A\times A:a+b=n\}.$ The Erdos--Turan conjecture asserts that, if $r(n) > 0$ for all sufficiently large $n$, then $r(n)$ is unbounded. Konstantoulas proved a density-theoretic version: if the upper density of $E=N\setminus(A+A)$ is less than $1/10$, then $\limsup_{n\to\infty} r(n)> 5$. In this paper, we improve Konstantoulas' constant to $7/32$. We also prove that $D(E)< 1/2$ implies $\limsup_{n\to\infty} r(n) > 3$, and give a conditional criterion forcing $\limsup_{n\to\infty} r(n)>7$. |
| title | An Improvement of Konstantoulas' Density Constant |
| topic | Number Theory |
| url | https://arxiv.org/abs/2605.30922 |