First comment is that this is an advanced research paper by a professional mathematician (from Poland, where they take their mathematics pretty seriously). You'll be doing very well indeed if you can make sense of all 20 pages of it.

To answer your specific questions:

1.surjective(in the context of the paper, which is talking about representations of numbers in some base) means that every number has a representation. (So for example, in the real numbers to the usual base 10, every number has a decimal representation, such as .)

2.pseudoinjectivemeans that most, but not quite all, numbers have a unique representation. (For example, 0.999... (recurring) is the same as 1.000..., so for the number 1 the representation is not unique.)

3. means "there exists", means "contained in", means "for all", means "x is an element of the set S" and is another way of saying the same thing. In this paper, means the 2-dimensional measure of the set S (measurebeing a generalisation of the concept of area). So the sentence " has at most one representation" means "There is a very small* set S of complex numbers with the property that for all complex numbers x except those in S, x has at most one representation."

* More precisely "very small" means "having measure zero".

In the first part of the paper, the shaded boxes represent the complex numbers (with integer real and imaginary parts) that can be represented in the form , where z (the base) is a given complex number, and the coefficients are integers between 0 and n-1, for some given n, and for some number k (n=2 and k=5 in the first two figures). I haven't tried to read beyond the first couple of pages, so I don't know what the rest of the paper contains (but it has some nice fractal pictures).