We study intermediate-scale statistics for the fractional parts of the sequence $\left(\alpha a_{n}\right)_{n=1}^{\infty}$, where $\left(a_{n}\right)_{n=1}^{\infty}$ is a positive, real-valued lacunary sequence, and $\alpha\in\mathbb{R}$. In particular, we consider the number of elements $S_{N}\!\left(L,\alpha\right)$ in a random interval of length $L/N$, where $L=O\!\left(N^{1-\epsilon}\right)$, and show that its variance (the number variance) is asymptotic to L with high probability w.r.t. $\alpha$, which is in agreement with the statistics of uniform i.i.d. random points in the unit interval. In addition, we show that the same asymptotic holds almost surely in $\alpha\in\mathbb{R}$ when $L=O\!\left(N^{1/2-\epsilon}\right)$. For slowly growing L, we further prove a central limit theorem for $S_{N}\!\left(L,\alpha\right)$ which holds for almost all $\alpha\in\mathbb{R}$.

For fixed $\alpha \in [0,1]$ , consider the set $S_{\alpha ,N}$ of dilated squares $\alpha , 4\alpha , 9\alpha , \dots , N^2\alpha \, $ modulo $1$ . Rudnick and Sarnak conjectured that, for Lebesgue, almost all such $\alpha $ the gap-distribution of $S_{\alpha ,N}$ is consistent with the Poisson model (in the limit as N tends to infinity). In this paper, we prove a new estimate for the triple correlations associated with this problem, establishing an asymptotic expression for the third moment of the number of elements of $S_{\alpha ,N}$ in a random interval of length $L/N$ , provided that $L> N^{1/4+\varepsilon }$ . The threshold of $\tfrac {1}{4}$ is substantially smaller than the threshold of $\tfrac {1}{2}$ (which is the threshold that would be given by a naïve discrepancy estimate).

Unlike the theory of pair correlations, rather little is known about triple correlations of the dilations $(\alpha a_n \, \text {mod } 1)_{n=1}^{\infty } $ for a nonlacunary sequence $(a_n)_{n=1}^{\infty } $ of increasing integers. This is partially due to the fact that the second moment of the triple correlation function is difficult to control, and thus standard techniques involving variance bounds are not applicable. We circumvent this impasse by using an argument inspired by works of Rudnick, Sarnak, and Zaharescu, and Heath-Brown, which connects the triple correlation function to some modular counting problems.

In Appendix B, we comment on the relationship between discrepancy and correlation functions, answering a question of Steinerberger.

We obtain estimates on the uniform convergence rate of the Birkhoff average of a continuous observable over torus translations and affine skew product toral transformations. The convergence rate depends explicitly on the modulus of continuity of the observable and on the arithmetic properties of the frequency defining the transformation. Furthermore, we show that for the one-dimensional torus translation, these estimates are nearly optimal.

The discrepancy function measures the deviation of the empirical distribution of a point set in $[0,1]^{d}$ from the uniform distribution. In this paper, we study the classical discrepancy function with respect to the bounded mean oscillation and exponential Orlicz norms, as well as Sobolev, Besov and Triebel–Lizorkin norms with dominating mixed smoothness. We give sharp bounds for the discrepancy function under such norms with respect to infinite sequences.

The half-open real unit interval (0,1] is closed under the ordinary multiplication and its residuum. The corresponding infinite-valued propositional logic has as its equivalent algebraic semantics the equational class of cancellative hoops. Fixing a strong unit in a cancellative hoop—equivalently, in the enveloping lattice-ordered abelian group—amounts to fixing a gauge scale for falsity. In this paper we show that any strong unit in a finitely presented cancellative hoop H induces naturally (i.e., in a representation-independent way) an automorphism-invariant positive normalized linear functional on H. Since H is representable as a uniformly dense set of continuous functions on its maximal spectrum, such functionals—in this context usually called states—amount to automorphism-invariant finite Borel measures on the spectrum. Different choices for the unit may be algebraically unrelated (e.g., they may lie in different orbits under the automorphism group of H), but our second main result shows that the corresponding measures are always absolutely continuous w.r.t. each other, and provides an explicit expression for the reciprocal density.

Let ${{\left( {{n}_{k}} \right)}_{k\ge 1}}$ be an increasing sequence of positive integers, and let $f\left( x \right)$ be a real function satisfying

1

$$f\left( x+1 \right)=f\left( x \right),\int\limits_{0}^{1}{f\left( x \right)}dx=0,\text{Va}{{\text{r}}_{\left[ 0,1 \right]}}f<\infty $$

If ${{\lim }_{k\to \infty }}\frac{{{n}_{k+1}}}{{{n}_{k}}}=\infty $ the distribution of

2

$$\frac{\sum\nolimits_{k=1}^{N}{f\left( {{n}_{k}}x \right)}}{\sqrt{N}}$$

converges to a Gaussian distribution. In the case

$$1<\underset{k\to \infty }{\mathop{\lim \inf }}\,\,\frac{{{n}_{k+1}}}{{{n}_{k}}},\,\,\underset{k\to \infty }{\mathop{\text{lim}\,\text{sup}}}\,\,\frac{{{n}_{k+1}}}{{{n}_{k}}}<\infty$$

there is a complex interplay between the analytic properties of the function $f$, the number-theoretic properties of ${{\left( {{n}_{k}} \right)}_{k\ge 1}}$, and the limit distribution of (2).

In this paper we prove that any sequence ${{\left( {{n}_{k}} \right)}_{k\ge 1}}$ satisfying $\lim {{\sup }_{k\to \infty }}{{n}_{k+1}}/{{n}_{k}}=1$ contains a nontrivial subsequence ${{\left( {{m}_{k}} \right)}_{k\ge 1}}$ such that for any function satisfying (1) the distribution of

$$\frac{\sum\nolimits_{k=1}^{N}{f\left( {{m}_{k}}x \right)}}{\sqrt{N}}$$

converges to a Gaussian distribution. This result is best possible: for any $\varepsilon >0$ there exists a sequence ${{\left( {{n}_{k}} \right)}_{k\ge 1}}$ satisfying lim $\underset{k\to \infty }{\mathop{\sup }}\,\frac{{{n}_{k+1}}}{{{n}_{k}}}\le 1+\varepsilon$ such that for every nontrivial subsequence ${{\left( {{m}_{k}} \right)}_{k\ge 1}}$ of ${{\left( {{n}_{k}} \right)}_{k\ge 1}}$ the distribution of (2) does not converge to a Gaussian distribution for some $f$.

Our result can be viewed as a Ramsey type result: a sufficiently dense increasing integer sequence contains a subsequence having a certain requested number-theoretic property.

A Diophantine m-tuple is a set A of m positive integers such that ab+1 is a perfect square for every pair a,b of distinct elements of A. We derive an asymptotic formula for the number of Diophantine quadruples whose elements are bounded by x. In doing so, we extend two existing tools in ways that might be of independent interest. The Erdős–Turán inequality bounds the discrepancy between the number of elements of a sequence that lie in a particular interval modulo 1 and the expected number; we establish a version of this inequality where the interval is allowed to vary. We also adapt an argument of Hooley on the equidistribution of solutions of polynomial congruences to handle reducible quadratic polynomials.

Let $p$ be an odd prime number, $K$ a $p$-adic field of degree $r$ over ${{\mathbf{Q}}_{p}}$, $O$ the ring of integers in $K,\,B\,=\,\{{{\beta }_{1}},\ldots .{{\beta }_{r}}\}$ an integral basis of $K$ over ${{\mathbf{Q}}_{p}}$, $u$ a unit in $O$ and consider sets of the form $N\,=\,\{{{n}_{1}}{{\beta }_{1}}\,+\ldots +\,{{n}_{r}}{{\beta }_{r}}\,:\,1\,\le \,{{n}_{j}}\,\le \,{{N}_{j}},\,1\,\le \,j\,\le \,r\}$. We show under certain growth conditions that the pair correlation of $\{u{{z}^{2}}\,:\,z\,\in N\}$ becomes Poissonian.

Let Tβ be the β-transformation on [0, 1). When β is an integer Tβ is ergodic with respect to Lebesgue measure and almost all orbits {} are uniformly distributed. Here we consider the non-integer case, determine when Tα, Tβ have the same invariant measure and when (appropriately normalised) orbits are uniformly distributed.

Let P be an IP-set of integers namely for a certain sequence The main questions studied here are : (1) Under what conditions on (an) is Pα dense modulo 1 for every irrational α? (2) Under what conditions on (an) is Pα (considered as a sequence ordered in a way to be subsequently defined) uniformly distributed modulo 1 for every irrational α?