In a recent paper of Ellenberg, Oberlin, and Tao [The Kakeya set and maximal conjectures for algebraic varieties over finite fields. Mathematika 56 (2010), 1–25], the authors asked whether there are Besicovitch phenomena in ${ \mathbb{F} }_{q} \mathop{[[t] ] }\nolimits ^{n} $. In this paper, we answer their question in the affirmative by explicitly constructing a Kakeya set of measure zero in ${ \mathbb{F} }_{q} \mathop{[[t] ] }\nolimits ^{n} $. Furthermore, we prove that any Kakeya set in ${ \mathbb{F} }_{q} \mathop{[[t] ] }\nolimits ^{2} $ or ${ \mathbb{Z} }_{p}^{2} $ is of Minkowski dimension 2.

We generalise an argument of Leader, Russell, and Walters to show that almost all sets of $d+ 2$ points on the $(d- 1)$-sphere ${S}^{d- 1} $ are not contained in a transitive set in some ${\mathbf{R} }^{n} $.

For a hyperbolic surface, embedded eigenvalues of the Laplace operator are unstable and tend to dissolve into scattering poles i.e. become resonances. A sufficient dissolving condition was identified by Phillips–Sarnak and is elegantly expressed in Fermi’s golden rule. We prove formulas for higher approximations and obtain necessary and sufficient conditions for dissolving a cusp form with eigenfunction $u_j$ into a resonance. In the framework of perturbations in character varieties, we relate the result to the special values of the $L$-series $L(u_j\otimes F^n, s)$. This is the Rankin–Selberg convolution of $u_j$ with $F(z)^n$, where $F(z)$is the antiderivative of a weight two cusp form. In an example we show that the above-mentioned conditions force the embedded eigenvalue to become a resonance in a punctured neighborhood of the deformation space.

We study the distribution of the size of Selmer groups arising from a 2-isogeny and its dual 2-isogeny for quadratic twists of elliptic curves with a non-trivial $2$-torsion point over $\mathbb {Q}$. This complements the work [Xiong and Zaharescu, Distribution of Selmer groups of quadratic twists of a family of elliptic curves. Adv. Math.219 (2008), 523–553] which studied the same subject for elliptic curves with full 2-torsions over $\mathbb {Q}$ and generalizes [Feng and Xiong, On Selmer groups and Tate–Shafarevich groups for elliptic curves $y^2=x^3-n^3$. Mathematika 58 (2012), 236–274.] for the special elliptic curves $y^2=x^3-n^3$. It is shown that the 2-ranks of these groups all follow the same distribution and in particular, the mean value is $\sqrt {\frac {1}{2}\log \log X}$ for square-free positive integers $n \le X$ as $X \to \infty $.

Answering two questions of Beresnevich and Velani, we develop zero-one laws in both simultaneous and multiplicative Diophantine approximation. Our proofs mainly rely on a versatile Cassels–Gallagher type theorem and the cross fibering principle of Beresnevich, Haynes and Velani.

A small value estimate is a statement providing necessary conditions for the existence of certain sequences of non-zero polynomials with integer coefficients taking small values at points of an algebraic group. Such statements are desirable for applications to transcendental number theory to analyze the outcome of the construction of an auxiliary function. In this paper, we present a result of this type for the product $ \mathbb {G}_{\mathrm {a}}\times \mathbb {G}_{\mathrm {m}}$ whose underlying group of complex points is $\mathbb {C}\times \mathbb {C}^{*}$. It shows that if a certain sequence of non-zero polynomials in $ \mathbb {Z}[X_1,X_2]$ takes small values at a point $(\xi ,\eta )$ together with their first derivatives with respect to the invariant derivation $\partial /\partial X_1 + X_2 (\partial /\partial X_2)$, then both $\xi $ and $\eta $ are algebraic over $\mathbb {Q}$. The precise statement involves growth conditions on the degree and norm of these polynomials as well as on the absolute values of their derivatives. It improves on a direct application of Philippon’s criterion for algebraic independence and compares favorably with constructions coming from Dirichlet’s box principle.

We find the best asymptotic lower bounds for the coefficient of the leading term of the ${L}_{1} $ norm of the two-dimensional axis-parallel discrepancy that can be obtained by Roth’s orthogonal function method among a large class of test functions. We use methods of combinatorics, probability, and complex and harmonic analysis.

It is well known that the classical Gauss sum, normalized by the square-root number of terms, takes only finitely many values. If one restricts the range of summation to a subinterval, a much richer structure emerges. We prove a limit law for the value distribution of such incomplete Gauss sums. The limit distribution is given by the distribution of a certain family of periodic functions. Our results complement Oskolkov’s pointwise bounds for incomplete Gauss sums as well as the limit theorems for quadratic Weyl sums (theta sums) due to Jurkat and van Horne and the second author.

In his Tata Lecture Notes, Igusa conjectured the validity of a strong uniformity in the decay of complete exponential sums modulo powers of a prime number and determined by a homogeneous polynomial. This was proved for non-degenerate forms by Denef–Sperber and then by Cluckers for weighted homogeneous non-degenerate forms. In a recent preprint, Wright has proved this for degenerate binary forms. We give a different proof of Wright’s result that seems to be simpler and relies upon basic estimates for exponential sums mod $p$as well as a type of resolution of singularities with good reduction in the sense of Denef.

We obtain explicit bounds on the moments of character sums, refining estimates of Montgomery and Vaughan. As an application we obtain results on the distribution of the maximal magnitude of character sums normalized by the square root of the modulus, finding almost double exponential decay in the tail of this distribution.

We investigate the intersections of the curve $ \mathbb{R} \ni t\mapsto \zeta (\frac{1}{2} + \mathrm{i} t)$ with the real axis. We show unconditionally that the zeta function takes arbitrarily large positive and negative values on the critical line.

Let $\mathcal {O} \subset \mathbb {R}^d$ be a bounded domain of class $C^{1,1}$. In the Hilbert space $L_2(\mathcal {O};\mathbb {C}^n)$, we consider a matrix elliptic second order differential operator $\mathcal {A}_{D,\varepsilon }$ with the Dirichlet boundary condition. Here $\varepsilon \gt 0$ is the small parameter. The coefficients of the operator are periodic and depend on $\mathbf {x}/\varepsilon $. There are no regularity assumptions on the coefficients. A sharp order operator error estimate $\|\mathcal {A}_{D,\varepsilon }^{-1} - (\mathcal {A}_D^0)^{-1} \|_{L_2 \to L_2} \leq C \varepsilon $ is obtained. Here $\mathcal {A}^0_D$is the effective operator with constant coefficients and with the Dirichlet boundary condition.

For an arbitrary subset $X$ of a finite-dimensional real Banach space $E$, the ball intersection with parameter $\lambda \gt 0$ is defined as the intersection of all balls of radius $\lambda $ whose centers are in $X$. On the other hand, the intersection of all balls of radius $\lambda $ that contain $X$ is said to be the respective ball hull. We present new results on these two notions and use them to get new insights into complete sets and (pairs of) sets of constant width, e.g., their representation as vector sums of suitable ball intersections and ball hulls. Also in this framework, we give partial answers to the known question, in what finite-dimensional real Banach spaces any complete set is of constant width. For polyhedral norms we obtain characterizations of monotypic balls via constant width properties of pairs formed by the ball intersection and ball hull of the same bounded and non-empty set. Finally, we present some new results on Borsuk numbers of sets of constant width in normed spaces, closely related to (unique) completions of compact sets. For example, the lower estimate on Borsuk numbers of bodies of constant width due to Lenz is extended to arbitrary normed spaces. Furthermore, we also derive the Borsuk number of the normed space with maximum norm.

Generalizing a result (the case $k= 1$) due to M. A. Perles, we show that any polytopal upper bound sphere of odd dimension $2k+ 1$ belongs to the generalized Walkup class ${ \mathcal{K} }_{k} (2k+ 1)$, i.e., all its vertex links are $k$-stacked spheres. This is surprising since it is far from obvious that the vertex links of polytopal upper bound spheres should have any special combinatorial structure. It has been conjectured that for $d\not = 2k+ 1$, all $(k+ 1)$-neighborly members of the class ${ \mathcal{K} }_{k} (d)$ are tight. The result of this paper shows that the hypothesis $d\not = 2k+ 1$ is essential for every value of $k\geq 1$.

We construct dense Borel measurable subgroups of Lie groups of intermediate Hausdorff dimension. In particular, we generalize the Erdős–Volkmann construction [Additive Gruppen mit vorgegebener Hausdorffscher Dimension, J. Reine Angew. Math. 221 (1966), 203–208], showing that any nilpotent $\sigma $-compact Lie group $N$ admits dense Borel subgroups of arbitrary dimension between zero and $\dim N$. In algebraic groups defined over a finite extension of the rationals, using diophantine properties of algebraic numbers, we are also able to construct dense subgroups of arbitrary dimension, but the general case remains open. In particular, we raise the following question: does there exist a measurable proper subgroup of $ \mathbb{R} $ of positive Hausdorff dimension which is stable under multiplication by a transcendental number? Subgroups of nilpotent $p$-adic analytic groups are also discussed.