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Mar
6
awarded  Revival
Feb
12
comment Primes and Parity
(cont) If one could do (and efficiently construct) this for a collection of APs of size, say, $n^{1/100}$ then it would solve the problem. It might be possible to probabilistically show the existence of such as collection. That wouldn't solve the problem on its own, but it would reduce the question to one of pseudo-randmness and not number theory. On the other hand, my intuition here isn't great and it might well be possible that such a collection can't exist.
Feb
12
comment Primes and Parity
Gil: yes, that is what I was thinking. I'd like to find a small collection of APs such that at least one is guaranteed to intersect any set (with certain properties which we can prove about the primes; size/density might be sufficient) in a set of odd parity...
Feb
10
comment Primes and Parity
A somewhat different question related to getting around the $\sqrt{n}$ barrier is the following: First recall that the Polymath4 result allows one to compute the parity of primes in an arithmetic progression intersecting [n,2n] "efficiently" (in time $n^{1/2-\delta}$ for some small $\delta$). Consider a subset $A \subseteq [n]$ of density $n/\log(n)$. What is the smallest collection of arithmetic progressions such that at least one is guaranteed to intersect $A$ with odd parity? More generally, it would be nice to reduce the problem to something not explicitly involving primes.
Jan
2
revised Connection between the Fourier transform of f and |f|
edited body
Jan
2
answered Connection between the Fourier transform of f and |f|
Nov
26
awarded  Enlightened
Nov
26
awarded  Nice Answer
Nov
26
awarded  Enlightened
Nov
26
awarded  Nice Answer
Nov
17
awarded  Good Question
Nov
13
awarded  Good Answer
Nov
7
answered A.e. pointwise convergence of L2 functions - counterexample for generalization of Carleson's thm
Oct
18
awarded  Yearling
Oct
13
answered $L^2$ boundedness of the Hilbert transform via Cotlar-Stein Lemma
Jul
6
answered Proof of the Friedlander–Iwaniec theorem
Jul
2
awarded  Curious
May
5
comment Distribution of $a^2+\alpha b^2$
This isn't an answer, but there is a theorem of Atkin which is in a similar spirit: These exists a sequence of integers such that the n-th term is $n^2 + O(log(n))$ and whose sumset has positive density in the integers. ams.org/mathscinet-getitem?mr=202687
Apr
29
answered A question on Cramer's theorem
Apr
9
comment Fourier series of functions on compact groups
@Anton, If you interpret the question like that it isn't very interesting. There are certainly $L^2$ functions on the circle without absolutely convergent Fourier series. You could alternately ask for unconditional pointwise convergence (that is a.e. pointwise convergence for every ordering) but this is false for every complete infinite orthonormal system (such as Fourier series on the cirlce, again).