Binary and ternary congruences involving intervals and sets modulo a prime
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866912577182760960 |
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| author | Garaev, Moubariz Z. Pardo, Julio C. Shparlinski, Igor E. |
| author_facet | Garaev, Moubariz Z. Pardo, Julio C. Shparlinski, Igor E. |
| contents | Let $s$ be a fixed positive integer constant, $\varepsilon$ be a fixed small positive number. Then, provided that a prime $p$ is large enough, we prove that for any set $\{{\mathcal M}\subseteq \mathbb F_p^*$ of size $|{\mathcal M}|= \lfloor p^{14/29}\rfloor$ and integer $H=\lfloor p^{14/29+\varepsilon}\rfloor$, any integer $λ$ can be represented in the form $$ \frac{m_1}{x_1^s}+\frac{m_2}{x_2^s}+\frac{m_3}{x_3^s}\equiv λ\bmod p, $$ with $$
m_i\in {\mathcal M}, \quad 1\le x_i\le H, \qquad i=1,2,3. $$ When $s=1$ we show that for almost all primes $p$ the following holds: if $|{\mathcal M}|= \lfloor p^{1/2}\rfloor$ and $H=\lfloor p^{1/2}(\log p)^{6+\varepsilon}\rfloor$, then any integer $λ$ can be represented in the form $$ \frac{m_1}{x_1}+\frac{m_2}{x_2}\equiv λ\bmod p, $$ with $$
m_i\in {\mathcal M}, \quad 1\le x_i\le H, \qquad i=1,2. $$ |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2410_03991 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Binary and ternary congruences involving intervals and sets modulo a prime Garaev, Moubariz Z. Pardo, Julio C. Shparlinski, Igor E. Number Theory Let $s$ be a fixed positive integer constant, $\varepsilon$ be a fixed small positive number. Then, provided that a prime $p$ is large enough, we prove that for any set $\{{\mathcal M}\subseteq \mathbb F_p^*$ of size $|{\mathcal M}|= \lfloor p^{14/29}\rfloor$ and integer $H=\lfloor p^{14/29+\varepsilon}\rfloor$, any integer $λ$ can be represented in the form $$ \frac{m_1}{x_1^s}+\frac{m_2}{x_2^s}+\frac{m_3}{x_3^s}\equiv λ\bmod p, $$ with $$ m_i\in {\mathcal M}, \quad 1\le x_i\le H, \qquad i=1,2,3. $$ When $s=1$ we show that for almost all primes $p$ the following holds: if $|{\mathcal M}|= \lfloor p^{1/2}\rfloor$ and $H=\lfloor p^{1/2}(\log p)^{6+\varepsilon}\rfloor$, then any integer $λ$ can be represented in the form $$ \frac{m_1}{x_1}+\frac{m_2}{x_2}\equiv λ\bmod p, $$ with $$ m_i\in {\mathcal M}, \quad 1\le x_i\le H, \qquad i=1,2. $$ |
| title | Binary and ternary congruences involving intervals and sets modulo a prime |
| topic | Number Theory |
| url | https://arxiv.org/abs/2410.03991 |