Explicit results on exponential sums
====================================

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   .. toctree::
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Collecting references:
:cite:`Daboussi-Rivat01`.



 
 



 
1. Bounds with the first derivative
-----------------------------------


We start with the Kusmin-Landau Lemma.

 
.. admonition:: Theorem
   :class: thm-tme-emt

   Let :math:`f` be a function over :math:`[a, b]` such that :math:`f^\prime` is monotonic
   and satisfies :math:`\theta \le f^\prime(u)\le 1-\theta` for some
   :math:`\theta \in(0,1/2]`. Then
   :math:`\displaystyle   \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
  :cite:`Cheng-Graham01`,
  see Lemma 2.3 of
  :cite:`Patel22`.

 
.. admonition:: Theorem (2004)
   :class: thm-tme-emt

   Let :math:`f` be a real-valued function with two continuous derivatives
   on :math:`[N+1, N+L]`. Suppose there are :math:`W > 1` and :math:`\lambda > 1` such
   that :math:`1 \le W |f^{\prime\prime}(x)| \le \lambda` for every
   :math:`x\in [N+1, N+L]`. Then we have
   :math:`\displaystyle   \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
  :cite:`Hiary16b`.
  See Lemma 2.3
  :cite:`Patel22`.


 
.. admonition:: Theorem (2016)
   :class: thm-tme-emt

   Let :math:`f` be a real-valued function with three continuous derivatives
   on :math:`[N+1, N+L]`. Suppose there are :math:`W > 1` and :math:`\lambda > 1` such
   that :math:`1 \le W |f^{\prime\prime\prime}(x)| \le \lambda` for every $x\in [N+1,
   N+L]:math:`. Then, for any `\eta > 0$, we have
   :math:`\displaystyle    \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
   :math:`\displaystyle   \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
   :math:`\displaystyle    \beta = \frac{65}{15\sqrt{\eta}} + \frac{3}{W^{1/3}}.` 



  

 
4. Bounds with higher derivatives
---------------------------------


See Lemma 3.1 and 3.2 of :cite:`Patel22`.

  
 
5. Iterated Van der Corput Inequality
-------------------------------------


During the proof of Lemma 8.6 in 
  :cite:`Granville-Ramare96`
  one finds the next inequality.

 
.. admonition:: Theorem (1996)
   :class: thm-tme-emt

   Let :math:`f` be a real-valued function with :math:`k+1` continuous derivatives
   on :math:`(A, B]` and let :math:`N` be a lower bound for the number of integers
   in :math:`(A,B]`. The quantity
   :math:`\displaystyle    \biggl|\frac{1}{8N}  \sum_{A < n\le B} \exp(2 i \pi f(n))\biggr|^{2^k}` 
   is bounded above by
   :math:`\displaystyle  \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 :math:`f_{r_1,\cdots,r_k}` satisfies
   :math:`\displaystyle  \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 :cite:`Karatsuba-Korolev07`.

 
.. admonition:: Theorem (2007)
   :class: thm-tme-emt

   Suppose :math:`f^\prime` is decreasing on :math:`[N+1,N+L]` and set
   :math:`f^\prime(N+L)=\alpha` and :math:`f^\prime(N) = \beta`.
   For integer :math:`\nu\in(\alpha, \beta]`, let :math:`x_\nu` be the solution
   to :math:`f^\prime(x)=\nu`. Suppose further that :math:`\lambda_2\le |f^{\prime\prime}(x)|\le h_2\lambda_2` and :math:`\lambda_3\le |f^{\prime\prime\prime}(x)|\le h_3\lambda_3`. Then
   :math:`\displaystyle   \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
   :math:`\displaystyle    |\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.`