Modular Ternary Additive Problems with Irregular or Prime Numbers

Our initial problem is to represent classes $m$ modulo $q$ by a sum of three summands, two being taken from rather small sets $\mathcal{A}$ and $\mathcal{B}$ and the third one having an odd number of prime factors (the so-called irregular numbers by S. Ramanujan) and lying in a $[q^{20r}, q^{20r}+q^{16r}]$ for some given $r\ge1$. We show that it is always possible to do so provided that $|\mathcal{A}||\mathcal{B}|\ge q(\log q)^2$. This proof leads us to study the trigonometric polynomial over irregular numbers in a short interval and to seek very sharp bound for them. We prove in particular that $\sum_{q^{20r}\le s \le q^{20r}+q^{16r}}e(sa/q)\ll q^{16r}(\log q)/\sqrt{\varphi(q)}$ uniformly in $r$, where $s$ ranges through the irregular numbers. We develop a technique initiated by Selberg and Motohashi to do so. In short, we express the characteristic function of the irregular numbers via a family of bilinear decomposition akin to Iwaniec amplification process and that uses pseudo-characters or local models. The technique applies to the Liouville function, to the Moebius function and also to the van Mangold function in which case it is slightly more difficult. It is however simple enough to warrant explicit estimates and we prove for instance that $| \sum_{X<\ell\le 2X}\Lambda(\ell)\, e(\ell a/q) |\le 1300 \sqrt{q}\,X/\varphi(q)$ for $250\le q\le X^{1/24}$ and any $a$ prime to $q$. Several other results are also proved.