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I would like to know what the current best estimation for the upper bound of the exponential sum $$\left|\sum_{n=1}^N \exp \left(2 \pi i\alpha\left(x_0+x_1 n+\ldots+x_d n^d\right)\right)\right|=\left|\sum_{n=1}^N e(P(n)\alpha)\right|$$ that only require $\alpha$ to be an irrational number and fix a given $d\geq 2$ and a polynomial with integer coefficients $P(n)=x_0+x_1 n+\ldots+x_d n^d$, where $x_i\in \mathbb{N}$ and $x_d\neq0$, and I have not found any particularly good references.

From the work of Weyl and Van der Corput, we know

$\left|\sum_{n=1}^N e(P(n)\alpha)\right| \lesssim_{d, \epsilon, \alpha} N^{1+\epsilon}\left(q^{-1}+N^{-1}+q N^{-k}\right)^\frac{1}{2^d}$

From the work of Vinogrodov, we know

$\left|\sum_{n=1}^N e(P(n)\alpha)\right| \lesssim_{d, \epsilon, \alpha} N^{1+\epsilon}\left(q^{-1}+N^{-1}+q N^{-k}\right)^\frac{1}{C d^2 \log d}$

From the work of Decoupling (J Bourgain, C Demeter, L Guth), we know

$\left|\sum_{n=1}^N e(P(n)\alpha)\right| \lesssim_{d, \epsilon, \alpha} N^{1+\epsilon}\left(q^{-1}+N^{-1}+q N^{-k}\right)^\frac{1}{d(d-1)}$

Thank you in advance.

I would like to know what the current best estimation for the upper bound of the exponential sum $$\left|\sum_{n=1}^N \exp \left(2 \pi i\alpha\left(x_0+x_1 n+\ldots+x_d n^d\right)\right)\right|=\left|\sum_{n=1}^N e(P(n)\alpha)\right|$$ that only require $\alpha$ to be an irrational number and fix a given $d\geq 2$ and a polynomial with integer coefficients $P(n)=x_0+x_1 n+\ldots+x_d n^d$, where $x_i\in \mathbb{N}$ and $x_d\neq0$, and I have not found any particularly good references.

From the work of Weyl and Van der Corput, we know

$\left|\sum_{n=1}^N e(P(n)\alpha)\right| \lesssim_{d, \epsilon, \alpha} N^{1+\epsilon}\left(q^{-1}+N^{-1}+q N^{-d}\right)^\frac{1}{2^d}$

From the work of Vinogrodov, we know

$\left|\sum_{n=1}^N e(P(n)\alpha)\right| \lesssim_{d, \epsilon, \alpha} N^{1+\epsilon}\left(q^{-1}+N^{-1}+q N^{-d}\right)^\frac{1}{C d^2 \log d}$

From the work of Decoupling (J Bourgain, C Demeter, L Guth), we know

$\left|\sum_{n=1}^N e(P(n)\alpha)\right| \lesssim_{d, \epsilon, \alpha} N^{1+\epsilon}\left(q^{-1}+N^{-1}+q N^{-d}\right)^\frac{1}{d(d-1)}$

Thank you in advance.

I would like to know what the current best estimation for the upper bound of the exponential sum $$\left|\sum_{n=1}^N \exp \left(2 \pi i\alpha\left(x_0+x_1 n+\ldots+x_d n^d\right)\right)\right|=\left|\sum_{n=1}^N e(P(n)\alpha)\right|$$ that only require $\alpha$ to be an irrational number and fix a given $d\geq 2$ and a polynomial with integer coefficients $P(n)=x_0+x_1 n+\ldots+x_d n^d$, where $x_i\in \mathbb{N}$ and $x_d\neq0$, and I have not found any particularly good references.

From the work of Weyl and Van der Corput, we know

$\left|\sum_{n=1}^N e(P(n)\alpha)\right| \lesssim_{d, \epsilon, \alpha} N^{1-\frac{1}{2^d}-\epsilon} $

Vinogrodov told us if irrational number $\alpha$ is Diophantine, then $ \left|\sum_{n=1}^N e(P(n)\alpha)\right| \lesssim_{d, \epsilon, \alpha} N^{1-\sigma(d)+\epsilon} \quad \text { where } \sigma(d) \geq \frac{1}{C d^2 \log d} $

Decoupling told us if irrational number $\alpha$ is Diophantine, then $\left|\sum_{n=1}^N e(P(n) \alpha)\right| \lesssim_{d, \epsilon, \alpha} N^{1-\sigma(d)+\epsilon} \quad$ where $\sigma(d) \geq \frac{1}{d(d-1)}$

Thank you in advance.

I would like to know what the current best estimation for the upper bound of the exponential sum $$\left|\sum_{n=1}^N \exp \left(2 \pi i\alpha\left(x_0+x_1 n+\ldots+x_d n^d\right)\right)\right|=\left|\sum_{n=1}^N e(P(n)\alpha)\right|$$ that only require $\alpha$ to be an irrational number and fix a given $d\geq 2$ and a polynomial with integer coefficients $P(n)=x_0+x_1 n+\ldots+x_d n^d$, where $x_i\in \mathbb{N}$ and $x_d\neq0$, and I have not found any particularly good references.

From the work of Weyl and Van der Corput, we know

$\left|\sum_{n=1}^N e(P(n)\alpha)\right| \lesssim_{d, \epsilon, \alpha} N^{1+\epsilon}\left(q^{-1}+N^{-1}+q N^{-k}\right)^\frac{1}{2^d}$

From the work of Vinogrodov, we know

$\left|\sum_{n=1}^N e(P(n)\alpha)\right| \lesssim_{d, \epsilon, \alpha} N^{1+\epsilon}\left(q^{-1}+N^{-1}+q N^{-k}\right)^\frac{1}{C d^2 \log d}$

From the work of Decoupling (J Bourgain, C Demeter, L Guth), we know

$\left|\sum_{n=1}^N e(P(n)\alpha)\right| \lesssim_{d, \epsilon, \alpha} N^{1+\epsilon}\left(q^{-1}+N^{-1}+q N^{-k}\right)^\frac{1}{d(d-1)}$

Thank you in advance.

I would like to know what the current best estimation for the upper bound of the exponential sum $$\left|\sum_{n=1}^N \exp \left(2 \pi i\alpha\left(x_0+x_1 n+\ldots+x_d n^d\right)\right)\right|=\left|\sum_{n=1}^N e(P(n)\alpha)\right|$$ that only require $\alpha$ to be an irrational number and fix a given $d\geq 2$ and a polynomial with integer coefficients $P(n)=x_0+x_1 n+\ldots+x_d n^d$, where $x_i\in \mathbb{N}$ and $x_d\neq0$, and I have not found any particularly good references.

From the work of Weyl and Van der Corput, we know

$\left|\sum_{n=1}^N e(P(n)\alpha)\right| \lesssim_{d, \epsilon, \alpha} N^{1-\frac{1}{2^k}-\epsilon} $

Vinogrodov told us if irrational number $\alpha$ is Diophantine, then $ \left|\sum_{n=1}^N e(P(n)\alpha)\right| \lesssim_{d, \epsilon, \alpha} N^{1-\sigma(k)+\epsilon} \quad \text { where } \sigma(d) \geq \frac{1}{C d^2 \log d} $

Thank you in advance.

I would like to know what the current best estimation for the upper bound of the exponential sum $$\left|\sum_{n=1}^N \exp \left(2 \pi i\alpha\left(x_0+x_1 n+\ldots+x_d n^d\right)\right)\right|=\left|\sum_{n=1}^N e(P(n)\alpha)\right|$$ that only require $\alpha$ to be an irrational number and fix a given $d\geq 2$ and a polynomial with integer coefficients $P(n)=x_0+x_1 n+\ldots+x_d n^d$, where $x_i\in \mathbb{N}$ and $x_d\neq0$, and I have not found any particularly good references.

From the work of Weyl and Van der Corput, we know

$\left|\sum_{n=1}^N e(P(n)\alpha)\right| \lesssim_{d, \epsilon, \alpha} N^{1-\frac{1}{2^d}-\epsilon} $

Vinogrodov told us if irrational number $\alpha$ is Diophantine, then $ \left|\sum_{n=1}^N e(P(n)\alpha)\right| \lesssim_{d, \epsilon, \alpha} N^{1-\sigma(d)+\epsilon} \quad \text { where } \sigma(d) \geq \frac{1}{C d^2 \log d} $

Decoupling told us if irrational number $\alpha$ is Diophantine, then $\left|\sum_{n=1}^N e(P(n) \alpha)\right| \lesssim_{d, \epsilon, \alpha} N^{1-\sigma(d)+\epsilon} \quad$ where $\sigma(d) \geq \frac{1}{d(d-1)}$

Thank you in advance.