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We obtain the exact value of the Hausdorff dimension of the set of coefficients of Gauss sums which for a given $alpha in (1/2,1)$ achieve the order at least $N^{alpha}$ for infinitely many sum lengths $N$. For Weyl sums with polynomials of degree $dge 3$ we obtain a new upper bound on the Hausdorff dimension of the set of polynomial coefficients corresponding to large values of Weyl sums. Our methods also work for monomial sums, match the previously known lower bounds, just giving exact value for the corresponding Hausdorff dimension when $alpha$ is close to $1$. We also obtain a nearly tight bound in a similar question with arbitrary integer sequences of polynomial growth.
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In this paper we prove the Hausdorff dimension of the set of (nondegenerate) singular two-dimensional vectors with uniform exponent $mu$ $in$ (1/2, 1) is 2(1 -- $mu$) when $mu$ $ge$ $sqrt$ 2/2, whereas for $mu$ textless{} $sqrt$ 2/2 it is greater than 2(1 -- $mu$) and at most (3 -- 2$mu$)(1 -- $mu$)/(1 + $mu$ + $mu$ 2). We also establish that this dimension tends to 4/3 (which is the dimension of the set of singular two-dimensional vectors) when $mu$ tends to 1/2. These results improve upon previous estimates of R. Baker, joint work of the first author with M. Laurent, and unpublished work of M. Laurent. We also prove a lower bound on the packing dimension that is strictly greater than the Hausdorff dimension for $mu$ $ge$ 0.565. .. .
We generalize the classical theorem by Jarnik and Besicovitch on the irrationality exponents of real numbers and Hausdorff dimension. Let a be any real number greater than or equal to 2 and let b be any non-negative real less than or equal to 2/a. We show that there is a Cantor-like set with Hausdorff dimension equal to b such that, with respect to its uniform measure, almost all real numbers have irrationality exponent equal to a. We give an analogous result relating the irrationality exponent and the effective Hausdorff dimension of individual real numbers. We prove that there is a Cantor-like set such that, with respect to its uniform measure, almost all elements in the set have effective Hausdorff dimension equal to b and irrationality exponent equal to a. In each case, we obtain the desired set as a distinguished path in a tree of Cantor sets.
Dobinski set $mathcal{D}$ is an exceptional set for a certain infinite product identity, whose points are characterized as having exceedingly good approximations by dyadic rationals. We study the Hausdorff dimension and logarithmic measure of $mathcal{D}$ by means of the Mass Transference Principle and by the construction of certain appropriate Cantor-like sets, termed willow sets, contained in $mathcal{D}$.
The purpose of this note is to record a consequence, for general metric spaces, of a recent result of David Bate. We prove the following fact: Let $X$ be a compact metric space of topological dimension $n$. Suppose that the $n$-dimensional Hausdorff measure of $X$, $mathcal H^n(X)$, is finite. Suppose further that the lower n-density of the measure $mathcal H^n$ is positive, $mathcal H^n$-almost everywhere in $X$. Then $X$ contains an $n$-rectifiable subset of positive $mathcal H^n$-measure. Moreover, the assumption on the lower density is unnecessary if one uses recently announced results of Csornyei-Jones.
We prove that there exist positive constants $C$ and $c$ such that for any integer $d ge 2$ the set of ${mathbf x}in [0,1)^d$ satisfying $$ cN^{1/2}le left|sum^N_{n=1}expleft (2 pi i left (x_1n+ldots+x_d n^dright)right) right|le C N^{1/2}$$ for infinitely many natural numbers $N$ is of full Lebesque measure. This substantially improves the previous results where similar sets have been measured in terms of the Hausdorff dimension. We also obtain similar bounds for exponential sums with monomials $xn^d$ when $d eq 4$. Finally, we obtain lower bounds for the Hausdorff dimension of large values of general exponential polynomials.
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