Mathematika最新文献

We prove that product-free subsets of the free group over a finite alphabet have maximum upper density $$ with respect to the natural measure that assigns total weight one to each set of irreducible words of a given length. This confirms a conjecture of Leader, Letzter, Narayanan, and Walters. In more general terms, we actually prove that strongly $$-product-free sets have maximum upper density $$ in terms of this measure. The bounds are tight.

We introduce the centred and the uncentred triangular maximal operators $$ and $$, respectively, on any locally finite tree in which each vertex has at least three neighbours. We prove that both $$ and $$ are bounded on $$ for every $$ in $$, that $$ is also bounded on $$, and that $$ is not of weak type (1, 1) on homogeneous trees. Our proof of the $$ boundedness of $$ hinges on the geometric approach of Córdoba and Fefferman. We also establish $$ bounds for some related maximal operators. Our results are in sharp contrast with the fact that the centred and the uncentred Hardy–Littlewood maximal operators (on balls) may be unbounded on $$ for every $$ even on some trees where the number of neighbours is uniformly bounded.

Let $$ be a $$-tuple of positive real numbers such that $$ and $$. A $$-dimensional vector $$ is said to be $$-singular if for every $$, there exists $$ such that for all $$, the system of inequalities

Let $$ be a subharmonic function on the open unit disc $$, centered at the origin of the complex plane, and let $$ be a holomorphic function such that $$. A classical result, known as Littlewood subordination principle, states $$, where $$ and $$ are integral means over the circle of radius $$ centered at the origin, of the functions $$ and $$, respectively. In this note, we obtain an unexpected improvement of Littlewood subordination principle in the case when the function $$ is univalent, by proving that

Let $$. We prove an unconditional lower bound on the measure of the sets $$ for $$. For $$, our bound has a Gaussian shape with variance proportional to $$. At the endpoint, $$, our result implies the best known $$-theorem for $$ that is due to Tsang. We also explain how the method breaks down for $$ given our current knowledge about the zeros of the zeta function. Conditionally on the Riemann hypothesis, we extend our results to the range $$.

For the Lagrange spectrum and other applications, we determine the smallest accumulation point of binary sequences that are maximal in their shift orbits. This problem is trivial for the lexicographic order, and its solution is the fixed point of a substitution for the alternating lexicographic order. For orders defined by cylinders, we show that the solutions are $$-adic sequences, where $$ is a certain infinite set of substitutions that contains Sturmian morphisms. We also consider a similar problem for symmetric ternary shifts, which is applicable to the multiplicative version of the Markoff–Lagrange spectrum.

We prove multiple vector-valued and mixed-norm estimates for multilinear operators in $$, more precisely for multilinear operators $$ associated to a symbol singular along a $$-dimensional space and for multilinear variants of the Hardy-Littlewood maximal function. When the dimension $$, the input functions are not necessarily in $$ and can instead be elements of mixed-norm spaces $$.

Such a result has interesting consequences especially when $$ spaces are involved. Among these, we mention mixed-norm Loomis-Whitney-type inequalities for singular integrals, as well as the boundedness of multilinear operators associated to certain rational symbols. We also present examples of operators that are not susceptible to isotropic rescaling, which only satisfy “purely mixed-norm estimates” and no classical $$ estimates.

Relying on previous estimates implied by the helicoidal method, we also prove (non-mixed-norm) estimates for generic singular Brascamp-Lieb-type inequalities.

Let $$ be pairwise distinct primes. From a theorem of Kummer, each prime $$ can divide $$ at most $$ times. We show that, for all $$, if $$ are sufficiently large in terms of $$ and $$, then there exist infinitely many positive integers $$ such that each $$ divides $$ at most $$ times. We connect this result to a famous conjecture by Graham on whether there are infinitely many integers $$ such that $$ is coprime to 105.

Let $$ be the Fourier coefficients of an $$ Hecke–Maass cusp form $$ and $$ be those of an $$ Hecke holomorphic or Hecke–Maass cusp form $$. Let $$ and $$ be a sequence. We show that if $$ for some $$,

Determinantal point processes exhibit an inherent repulsive behavior, thus providing examples of very evenly distributed point sets on manifolds. In this paper, we study the so-called harmonic ensemble, defined in terms of Laplace eigenfunctions on the sphere $$ and the flat torus $$, and the so-called spherical ensemble on $$, which originates in random matrix theory. We extend results of Beltrán, Marzo, and Ortega-Cerdà on the Riesz $$-energy of the harmonic ensemble to the nonsingular regime $$, and as a corollary find the expected value of the spherical cap $$ discrepancy via the Stolarsky invariance principle. We find the expected value of the $$ discrepancy with respect to axis-parallel boxes and Euclidean balls of the harmonic ensemble on $$. We also show that the spherical ensemble and the harmonic ensemble on $$ and $$ with $$ points attain the optimal rate $$ in expectation in the Wasserstein metric $$, in contrast to independent and identically distributed random points, which are known to lose a factor of $$.