I'm attempting to create this spiraling series of circles for a project but I'm not entirely sure how to do it, they will serve as a border and background for a central object
Here is a different approach that can be easily animated directly in the viewport using objects.
Create a new circle mesh with a filled face. In Edit Mode assign all vertex to a new vertex group with a weight of $1$.
Add a bunch of Array Modifiers to create a dot grid with the desired spacing and density.
Now add add Vertex Weigh Proximity Modifier and make it control the vertex group created above. Adjust the values as necessary for your scene scale.
Now add a Corrective Smooth modifier to your arrayed circle, activate the Only Smooth option and increase the Iterations value as desired (in my case to about 200).
Pick the previous vertex group to affect its weight.
Now use any dummy object to control the smoothing effect with any desired shape. In this case I created a circular mesh, but can be any mesh shape, then assign this object to control the Vertex Weigh Proximity Modifier created above. Moving it about, scaling it or changing it's shape will affect the halftone effect.
Add a particle system. Set the emission frame range inbetween the first frame, e.g.
Set the particle emission settings, set Source > Distribution to Grid and increase the grids resolution until you have enough particles.
Under the render section of the particle settings set the Render as to Object and create and choose a circle object.
As a result you will have an even grid of circles. Turn off all movement for the particle system (like Initial Velocity or the Gravity field) making it static.
Now add a texture to the particle system under the textures section. Open that texture and choose a blend type texture. Make it spherical and control it's color falloff with a colorramp. Set it to influence the size of the particles. Using the colorramp we can also invert the spherical texture, making the center black, which results in the center particles getting scale to
0. Under the Mapping section, you can use the offset fields to position the texture.
To render it like your reference, use a tilted rotated camera with a short focal length and set the particles object's color (the circle material) to white and the background to black.