Hello,

I too would be very curious about this bound and will think further about this. The nice thing about the $u_i(y)$, as I recall my advisor having told me, is that the estimates for geometric sums can be independent of their length. So, similar to an estimate for geometric sums found in Chapter 25 of H. Davenport, Multiplicative Number Theory, Third Edition, one has that

$$
u_i(y) \ll \min\left(N, \frac{1}{\|b_iy\|}\right).
$$ 

Following the perspective of this enlightening post, <a href="http://terrytao.wordpress.com/2011/12/31/montgomerys-uncertainty-principle/">Montgomery's uncertainty principle</a>, from the blog of Professor Tao, another nice way to think of it is that the $u_i(y)$ is the Fourier transform of a function $f: \mathbb{Z} \rightarrow \{0,1\}$ that avoids $p-1$ residue classes modulo $p$ for each prime $p$ dividing $b_i$.

A technique for the estimate you require which initially occurred to me was one I learnt from:

S. Baier, L. Zhao, <a href="http://arxiv.org/abs/math/0605563">Primes in Quadratic Progressions on Average</a>, Math. Ann., Vol. 338, 2007, No. 4, pp. 963-982.

The analogous situation to the bound you need is treated there at equations (4.2) and (4.3). But there is a difference between that problem and this one. Over there, one only has the case $b_i=1$ and the maximum value of $\sum_{m\leq N}e(m y)$ can be avoided by avoiding $(-Q/N,Q/N)$. But here, when $b_i>1$, the interval $(Q/N, 1/2)$ still contains values where $u_i(y)$ attains its maximum.

So one observes that by naively applying Hölder's inequality with $\frac{1}{n_1}+\frac{1}{n_2}+\frac{1}{n_3}=1$ to obtain

$$
\int_{Q/N}^{1/2}\left|u_1(y)u_2(y)u_3(y)\right|dy \ll \prod_{i=1}^3\left(\int_{Q/N}^{1/2}\left|u_i(y)\right|^{n_i}dy\right)^{1/n_i},
$$

where, for each $u_i(y)$, if one ignores that one is integrating from $Q/N$ and instead integrates from $0$, one has that

$$
\int_{0}^{1/2}\left|u_i(y)\right|^{n_i}dy \ll b_i\left(\int_{0}^{1/(b_iN)}N^{n_i} dy + \int_{1/(b_iN)}^{1/2b_i}\frac{1}{(b_iy)^{n_i}}dy\right) \ll N^{n_i-1}
$$

which only gives $O(N^2)$ for the term you need to estimate. It can be observed that throwing away the part from $(0,Q/N)$ will not save much like this. Letting $b=\max(b_1,b_2,b_3)$ and applying Hölder's inequality to the interval $(Q/N,1/(2b))$ gives

$$
\int_{Q/N}^{1/(2b)}u_1(y)u_2(y)u_3(y)dy \ll \prod_{i=1}^3\left(\int_{Q/N}^{1/(2b)}\left|\frac{1}{b_iy}\right|^{n_i}dy\right)^{1/n_i} \ll \frac{1}{b_1b_2b_3}\left(\frac{N}{Q}\right)^2,
$$

but this is only the easy part of the interval to treat like this. 

Here are some special cases:

**Case 1** Amongst, $b_1,b_2,b_3$, at least one $b_i$=1. Without loss of generality, suppose $b_1=1$. Then by Cauchy's inequality and Parserval's identity, 

$$
\int_{Q/N}^{1/2}u_1(y)u_2(y)u_3(y)dy \ll \sup_{y\in (Q/N,1/2)}u_1(y)\left(\int_0^1\left|u_2(y)\right|^2dy\right)^{1/2} \left(\int_0^1\left|u_3(y)\right|^2dy\right)^{1/2} 
$$

which is $O\left(N^2/Q\right)$.

**Case 2** $b_1,b_2,b_3$ are pairwise relatively prime and $|b_i|<R$, say, for $R$ sufficiently small. The interval $(Q/N,1/(2b))$ was dealt with, so here we consider $(1/(2b),1/2)$. For $i=1,2,3$, let 

$$
\mathfrak{M}_{b_i} = \bigcup_{k=0}^{b_i-1}\left[\frac{k}{b_i}-\frac{1}{b_iN},\frac{k}{b_i}+\frac{1}{b_iN}\right].
$$ 

Then the conditions for this case imply that the $\mathfrak{M}_{b_i}$ only intersect each other at an interval centered at zero that is contained in $(-1/(2b),1/(2b))$. The idea is that the $u_i(y)$ are not simultaneously large on $(1/(2b),1-1/(2b))$. Without loss of generality, consider $u_1(y)$ and $u_2(y)$ on $(1/(2b),1-1/(2b))$ and suppose that $u_1(y) \gg N^{1/2}$. Then $y$ is of the form

$$
y= \frac{k \pm \beta}{b_1},
$$

$k \not=0$, $\beta>0$ and $1/\beta > N^{1/2}$. But then, 

$$
\frac{1}{\|b_2y\|} = \frac{b_1}{(kb_2 \bmod b_1) + \beta} = O(b_1)
$$

since $k\not=0$. Therefore, this implies that on $(1/(2b),1/2)$, one has that $|u_1u_2u_3|\ll NR^2$.

**References:**

There are some papers that deal with this kind of problem using the circle method. They are:

A. Balog, Linear Equations in Primes, Mathematika (1992), 39 : pp 367-378 

M.C. Liu, K.M. Tsang, Small prime solutions of linear equations, Théorie des nombres (Quebec, PQ, 1987), 595–624, de Gruyter, Berlin, 1989.

M.C. Liu, K.M. Tsang, Small prime solutions of some additive equations, Monatsh. Math. 111 (1991), no. 2, 147–169.