Modular Ternary Additive Problems with Irregular or Prime Numbers

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