Halftoning is a time-honored printing technique that simulates continuous tones using ink dots (halftone dots). The resurgence of deep learning has catalyzed the emergence of innovative technologies in the printing industry, fostering the advancement of data-driven halftoning methods. Nevertheless, current deep learning-based approaches produce halftones through image-to-image black box transformations, lacking direct control over the movement of individual halftone dots. In this paper, we propose an innovative halftoning method termed ``neural dot-controllable halftoning". This method allows dot-level image dithering by providing direct control over the motion of each ink dot. We conceptualize halftoning as the process of sprinkling dots on a canvas. Initially, a specific quantity of dots are randomly dispersed on the canvas and subsequently adjusted based on the surrounding grayscale and gradient. To establish differentiable transformations between discrete ink dot positions and halftone matrices, we devise a lightweight dot encoding network to spread dense gradients to sparse dots. Dot control offers several advantages to our approach, including the capability to regulate the quantity of halftone dots and enhance specific areas with artifacts in the generated halftones by adjusting the placement of the dots. Our proposed method exhibits superior performance than previous approaches in extensive quantitative and qualitative experiments.