Unsolved problems

Here are some problems I tried but could not solve. They reflect only my personal taste and (lack of) mathematical abilities. I tried to avoid well-known questions here, so, despite I spent a lot of time on, say, the two-weight problem for the Hilbert transform (with very limited success), I'm not putting the corresponding question here: it is very well known without my advertising it on my web page. At last, keep in mind that this page is permanently under construction. I will keep adding to it whenever I have free time to sit and play with HTML. Also, I apologize for the clumsy sentences and hard-to-read formulae: current browsers do not know TeX. I thought of making everything into a PDF file but then I decided against it in favor of making this page more interactive...

If you get interested in and can make some progress on any of these questions, I would be happy to hear from you.

1) Set of positive values of a harmonic polynomial cannot be too small

Let P be a harmonic polynomial of degree N>0 in R³ such that P(0)>0. Is it true that, for every d>0, there exists c_d>0 such that the measure of the set of points x in the unit ball satisfying P(x)>0 is at least c_dN^-d?

2) Divide a triangle into congruent pieces

For which integer n is it possible to divide an equilateral triangle into n congruent polygons (not necessarily connected)?

3) Almost periodic sequence of analytic functions

Suppose that the functions f_k are analytic in the unit disk and continuous up to the boundary. Suppose also that, for each z on the unit circumference, the sequence f_k(z) is almost periodic. Does it follow that the sequence f_k(0) is almost periodic?

4) Salem inequality in L^p

Let 0<p<2. Let f be a function on the unit circumference T whose spectrum is sparce in the sense that the difference between any two its elements is greater than some very large number N. Is it true that the integral of |f|^p over any sufficiently small arc is much smaller than the integral over the entire circumference?

5) Polynomials with random sparce spectrum

Let 1/2<p<1. Let I=[0,2π]. Let A be a random subset of the set 1,2,...,n with n^p elements. Is it true that there exists a positive function F independent of n such that, with probability 0.99, for any trigonometric polynomial P with spectrum A and for any measurable subset E of I of positive measure, the integral of |P|² over E is at least F(|E|) times the integral of |P|² over I?

6) Variant of Turan's lemma

Let I=[0,1]. Let E be a measurable subset of I of very small measure. Does there exist a function B(x) tending to 0 as x tends to 0 such that for any exponential polynomial P(t)=∑_1≤k≤nexp(iρ_kt), which is a linear combination of n arbitrary purely imaginary exponents (ρ_kbelong to R), the maximum of |P| over the complement of E in I is at least e^{-B(|E|) n+o(n)} times the maximum over the entire I?

7) L^∞-bound for an energy-preserving chain

Consider the infinite chain of differential equations du₀/dt=f(t)-u₀u₁, du_j/dt=u_j-1²-u_ju_j+1 (j≥1). Suppose that at the initial moment 0≤u_j≤1 for all j and 0≤f(t)≤1 for all t. Can one conclude that u_j(t)≤100, say, for all j and for all t.

8) Sets of convergence of Taylor series

Let c_k be a sequence of complex numbers converging to 0. Consider the Taylor series ∑_k≥0c_kz^k. Let A be the set of all z on the unit circumference for which this series converges. If is clear that A is an F_σδ-set. The question is whether every F_σδ-set is a set of convergence for some Taylor series. The best known result (which is fairly easy) is that every F_σ-set is a set of convergence.

9) Fancy maximal function

For a non-negative integrable function f, denote by A(f,x,h) the average of f over the interval (x-h,x+h). Let λ₁, λ₂, λ₃,... be a sequence of positive numbers. Let f₁,f₂,f₃,... be a sequence of non-negative integrable functions. Consider the maximal function M(x)=sup_t>0∑_k≥1A(f_k,x,λ_kt) . Is it true that the measure of the set where M(x)>y does not exceed some absolute constant times the sum of the integrals of f_k divided by y?

10) Joel Shapiro's question about an analytic function in the unit disk

Let f be an analytic function in the unit disk such that |f(z)|≤1 for all z in the unit disk and f is not the identity function. Is it true that the integrals of f(z)ⁿz^-n over the unit circumference with respect to the Lebesgue measure tend to 0 as n tends to infinity?

11) Product of two integrable functions with disjoint spectra

Let f and g be two integrable functions on the unit circumference such that |f||g| is integrable and the spectra of f and g are disjoint. Is it true that the integral of f times g-bar over the unit circumference with respect to the Lebesgue measure must be 0?

More later...