Integral Concentration of idempotent trigonometric polynomials with gaps

We prove that for all p>1/2 there exists a constant $\gamma_p>0$ such that, for any symmetric measurable set of positive measure E\subset \TT and for any $\gamma<\gamma_p$, there is an idempotent trigonometrical polynomial f satisfying \int_E |f|^p > \gamma \int_{\TT} |f|^p. This disproves a conjecture of Anderson, Ash, Jones, Rider and Saffari, who proved the existence of $\gamma_p>0$ for p>1 and conjectured that it does not exists for p=1. Furthermore, we prove that one can take $\gamma_p=1$ when p>1 is not an even integer, and that polynomials f can be chosen with arbitrarily large gaps when $p\neq 2$. This shows striking differences with the case p=2, for which the best constant is strictly smaller than 1/2, as it has been known for twenty years, and for which having arbitrarily large gaps with such concentration of the integral is not possible, according to a classical theorem of Wiener. We find sharper results for $0<p\leq 1$ when we restrict to open sets, or when we enlarge the class of idempotent trigonometric polynomials to all positive definite ones.Comment: 43 pages; to appear in Amer. J. Mat

Distribution of Beurling primes and zeroes of the Beurling zeta function I. Distribution of the zeroes of the zeta function of Beurling

We prove three results on the density resp. local density and clustering of zeros of the Beurling zeta function $\zeta(s)$ close to the one-line $\sigma:=\Re s=1$. The analysis here brings about some news, sometimes even for the classical case of the Riemann zeta function. Theorem 4 provides a zero density estimate, which is a complement to known results for the Selberg class. Note that density results for the Selberg class rely on use of the functional equation of $\zeta$, which we do not assume in the Beurling context. In Theorem 5 we deduce a variant of a well-known theorem of Tur\'an, extending its range of validity even for rectangles of height only $h=2$. In Theorem 6 we will extend a zero clustering result of Ramachandra from the Riemann zeta case. A weaker result -- which, on the other hand, is a strong sharpening of the average result from the classic book \cite{Mont} of Montgomery -- was worked out by Diamond, Montgomery and Vorhauer. Here we show that the obscure technicalities of the Ramachandra paper (like a polynomial with coefficients like $10^8$) can be gotten rid of, providing a more transparent proof of the validity of this clustering phenomenon

A potential theoretic minimax problem on the torus

We investigate an extension of an equilibrium-type result, conjectured by Ambrus, Ball and Erd\'elyi, and proved recently by Hardin, Kendall and Saff. These results were formulated on the torus, hence we also work on the torus, but one of the main motivations for our extension comes from an analogous setup on the unit interval, investigated earlier by Fenton. Basically, the problem is a minimax one, i.e. to minimize the maximum of a function $F$, defined as the sum of arbitrary translates of certain fixed "kernel functions", minimization understood with respect to the translates. If these kernels are assumed to be concave, having certain singularities or cusps at zero, then translates by $y_j$ will have singularities at $y_j$ (while in between these nodes the sum function still behaves realtively regularly). So one can consider the maxima $m_i$ on each subintervals between the nodes $y_j$, and look for the minimization of $\max F = \max_i m_i$. Here also a dual question of maximization of $\min_i m_i$ arises. This type of minimax problems were treated under some additional assumptions on the kernels. Also the problem is normalized so that $y_0=0$. In particular, Hardin, Kendall and Saff assumed that we have one single kernel $K$ on the torus or circle, and $F=\sum_{j=0}^n K(\cdot-y_j)= K + \sum_{j=1}^n K(\cdot-y_j)$. Fenton considered situations on the interval with two fixed kernels $J$ and $K$, also satisfying additional assumptions, and $F= J + \sum_{j=1}^n K(\cdot-y_j)$. Here we consider the situation (on the circle) when \emph{all the kernel functions can be different}, and $F=\sum_{j=0}^n K_j(\cdot- y_j) = K_0 + \sum_{j=1}^n K_j(\cdot-y_j)$. Also an emphasis is put on relaxing all other technical assumptions and give alternative, rather minimal variants of the set of conditions on the kernel