I have a question:

– page 163, exercise 4.3.4 : I understand how to prove it if the group is cyclic.

– does the fact, that the group has bilinear form that indues a isomorpism means, that this is always the case ( because the circle group is cyclic)?

all the best

Tomer

– you wrote that that P(A_1)***P(A_n) is the quantity one would

expect if random selection is involved conditioning on x =a_1 + .. + a_n,

restricted to these random sets.

– essentially meaning, that the representation function at a fixed point

is expected to be P(A_1)***P(A_n)*(|Z|^(n-1)) restricted to these sets.

the first part of P(A_1)***P(A_n) is understood, but the last part (|Z|^(n-1))

would describe the representation function at a fixes point where solutions

are unconditioned. I find it hard to believe and follow this claim.

please clarify, it is important for me to understand.

T

]]>*[Uncountable sets can still have zero measure. For instance, the set has zero two-dimensional Lebesgue measure for any given number . -T.]*

I am getting back to my thesis,

and was wondering about the solution of

exercise 2.1.1.

– how is it possible to come up with a uniform probability

function over the an infinite set of real numbers??

– what am I missing?

all the best

Tomer Shalev

*[The exercise is referring to Lebesgue measure on the unit interval [0,1], which is a probability measure. -T.]*

Yes. The definition of mathematical fields is very fuzzy, but there are many questions which seem to belong to both “Number Theory” and “Combinatorics”. For example: in how many ways can a positive integer be represented as a sum of four integer sq…

]]>On p239, exercise 5.5.17: I don’t think this example works. The problem is that the midpoints of the quadrilateral , , , themselves form a parallelogram, and this rules out any non-standard Freiman homomorphisms to .

Replacing the generic quadrilateral with a generic pentagon in should fix it, I think, giving a universal ambient group of .

(Also, I think there is a typo

on the penultimate line: the first “2” should be a “4”.)

*[Correction added, thanks – T.]*

On p241, proof of Lemma 5.45, last line of the first displayed equation: I think

should read

as an estimate on the binomial coefficient.

*[Correction added, thanks – T.]*