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Register a new userEstimates of the Discrepancy Function in Exponential Orlicz Spaces
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We prove that in all dimensions n at least 3, for every integer N there exists a distribution of points of cardinality $ N$, for which the associated discrepancy function D_N satisfies the estimate an estimate, of sharp growth rate in N, in the exponential Orlicz class exp)L^{2/(n+1)}. This has recently been proved by M.~Skriganov, using random digit shifts of binary digital nets, building upon the remarkable examples of W.L.~Chen and M.~Skriganov. Our approach, developed independently, complements that of Skriganov.
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Let A_N be an N-point distribution in the unit square in the Euclidean plane. We consider the Discrepancy function D_N(x) in two dimensions with respect to rectangles with lower left corner anchored at the origin and upper right corner at the point x. This is the difference between the actual number of points of A_N in such a rectangle and the expected number of points - N x_1x_2 - in the rectangle. We prove sharp estimates for the BMO norm and the exponential squared Orlicz norm of D_N(x). For example we show that necessarily ||D_N||_(expL^2) >c(logN)^(1/2) for some aboslute constant c>0. On the other hand we use a digit scrambled version of the van der Corput set to show that this bound is tight in the case N=2^n, for some positive integer n. These results unify the corresponding classical results of Roth and Schmidt in a sharp fashion.
In the current paper we obtain discrepancy estimates in exponential Orlicz and BMO spaces in arbitrary dimension $d ge 3$. In particular, we use dyadic harmonic analysis to prove that for the so-called digital nets of order $2$ the BMO${}^d$ and $exp big( L^{2/(d-1)} big)$ norms of the discrepancy function are bounded above by $(log N)^{frac{d-1}{2}}$. The latter bound has been recently conjectured in several papers and is consistent with the best known low-discrepancy constructions. Such estimates play an important role as an intermediate step between the well-understood $L_p$ bounds and the notorious open problem of finding the precise $L_infty$ asymptotics of the discrepancy function in higher dimensions, which is still elusive.
We investigate arithmetic properties of values of the entire function $$ F(z)=F_q(z;lambda)=sum_{n=0}^inftyfrac{z^n}{prod_{j=1}^n(q^j-lambda)}, qquad |q|>1, quad lambda otin q^{mathbb Z_{>0}}, $$ that includes as special cases the Tschakaloff function ($lambda=0$) and the $q$-exponential function ($lambda=1$). In particular, we prove the non-quadraticity of the numbers $F_q(alpha;lambda)$ for integral $q$, rational $lambda$ and $alpha otin-lambda q^{mathbb Z_{>0}}$, $alpha e0$.
In this paper, we prove some extensions of recent results given by Shkredov and Shparlinski on multiple character sums for some general families of polynomials over prime fields. The energies of polynomials in two and three variables are our main ingredients.
It is a well-known conjecture in the theory of irregularities of distribution that the L1 norm of the discrepancy function of an N-point set satisfies the same asymptotic lower bounds as its L^2 norm. In dimension d=2 this fact has been established by Halasz, while in higher dimensions the problem is wide open. In this note, we establish a series of dichotomy-type results which state that if the L^1 norm of the discrepancy function is too small (smaller than the conjectural bound), then the discrepancy function has to be large in some other function space.
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