The TME-EMT project
Explicit results on exponential sums
Collecting references:
.
1. Bounds with the first derivative
We start with the Kusmin-Landau Lemma.
Theorem
Let $f$ be a function over $[a, b]$ such that $f^\prime$ is monotonic
and satisfies $\theta \le f^\prime(u)\le 1-\theta$ for some $\theta
\in(0,1/2]$. Then
$$
\biggl|\sum_{a\le n\le b} e(f(n))\biggr|
\le
\cot\frac{\pi\theta}{2}\le \frac{2}{\pi \theta}.
$$
2. Bounds with the second derivative
Here is a corrected version of Lemma 3 of
,
see Lemma 2.3 of
.
Theorem (2004)
Let $f$ be a real-valued function with two continuous derivatives
on $[N+1, N+L]$. Suppose there are $W > 1$ and $\lambda > 1$ such
that $1 \le W |f^{\prime\prime}(x)| \le \lambda$ for every $x\in [N+1,
N+L]$. Then we have
$$
\biggl|\sum_{n= N+1}^{N+L}
\exp( 2i\pi f(n)) \biggr|
\le 2\biggl(\frac{L \lambda}{W} +2\biggr)
\biggl(2\sqrt{\frac{W}{\pi}} + 1\biggr).
$$
3. Bounds with the third derivative
Here is Lemma 1.2 of
.
See Lemma 2.3
.
Theorem (2016)
Let $f$ be a real-valued function with three continuous derivatives
on $[N+1, N+L]$. Suppose there are $W > 1$ and $\lambda > 1$ such
that $1 \le W |f^{\prime\prime\prime}(x)| \le \lambda$ for every $x\in [N+1,
N+L]$. Then, for any $\eta > 0$, we have
$$
\biggl|\sum_{n= N+1}^{N+L}
\exp( 2i\pi f(n)) \biggr|^2
\le (LW^{-1/3} +\eta) (\alpha L + \beta W^{2/3})
$$
where
$$
\alpha = \frac{1}{\eta} +\frac{64\lambda}{75}
\sqrt{\eta + W^{-1/3}}+\frac{\lambda\eta}{W^{1/3}}
+\frac{\lambda}{W^{2/3}},
$$
and
$$
\beta = \frac{65}{15\sqrt{\eta}} + \frac{3}{W^{1/3}}.
$$
4. Bounds with higher derivatives
See Lemma 3.1 and 3.2 of
.
5. Iterated Van der Corput Inequality
During the proof of Lemma 8.6 in
,
one finds the next inequality.
Theorem (1996)
Let $f$ be a real-valued function with $k+1$ continuous derivatives
on $(A, B]$ and let $N$ be a lower bound for the number of integers
in $(A,B]$. The quantity
$$
\biggl|\frac{1}{8N}
\sum_{A < n\le B} \exp(2 i \pi f(n))\biggr|^{2^k}
$$
is bounded above by
$$
\frac{1}{8}\biggl(
\frac{1}{Q} + \frac{1}{Q^{2-2^{1-k}}}
\sum_{r_1 =1}^{Q2^{-0}}
\sum_{r_2 =1}^{Q2^{-1}}
\cdots
\sum_{r_k =1}^{Q2^{-k+1}}
\biggl|
\frac{1}{N}
\sum_{A < n \le B-r_1-r_2-\cdots-r_k}
\exp(\pm 2i\pi f_{r_1,\cdots,r_k}(n))
\biggr|
\biggr)
$$
where the function $f_{r_1,\cdots,r_k}$ satisfies
$$
\forall t,\ \exists y\in[t, t + r_1 + \cdots + r_k],
\quad
f^{\prime}_{r_1,\cdots, r_k}(t) = r_1r_2\cdots r_k f^{(k+1)}(y).
$$
6. Explicit Poisson Formula
Here is a consequence of the main theorem of
.
Theorem (2007)
Suppose $f^\prime$ is decreasing on $[N+1,N+L]$ and set
$f^\prime(N+L)=\alpha$ and $f^\prime(N) = \beta$.
For integer $\nu\in(\alpha, \beta]$, let $x_\nu$ be the solution
to $f^\prime(x)=\nu$. Suppose further that
$\lambda_2\le |f^{\prime\prime}(x)|\le h_2\lambda_2$ and
$\lambda_3\le |f^{\prime\prime\prime}(x)|\le h_3\lambda_3$. Then
$$
\sum_{n=N+1}^{N+L} \exp(2i\pi f(n))
=
\sum_{\alpha < \nu\le \beta}
\frac{\exp(2i\pi (f(x_\nu)-\nu x_\nu-1/8))}{\sqrt{f^{\prime\prime}(x_\nu)}}
+\mathcal{E}
$$
where
$$
|\mathcal{E}|
\le \frac{40}{\sqrt{\pi}}\lambda^{-1/2}
+ \frac{3+2h_2}{\pi} \log(\beta-\alpha+2)
+ 2.9 h_2h_3^{1/5}L(\lambda_2\lambda_3)^{1/5}
+1.9.
$$
Last updated on July 12th, 2023, by Olivier Ramaré