Metric results on the discrepancy of sequences \((a_{n}{\alpha})_{n\geqslant 1}\) modulo one for integer sequences \((a_{n})_{n\geqslant 1}\) of polynomial growth (Q2810735)

The uniformly distributed sequences are a class of sequences with many applications in different branches of science. The discrepancy of a given sequence shows the quality of the distribution of the points of this sequence. In general, it is interesting to know the lower bounds which give the convergence rate to zero of the discrepancy.NEWLINENEWLINEIn the present paper, the distribution properties of sequences of the form \((\{a_n \alpha\})_{n \geq 1}\) in the unit interval, where \((a_n)_{n \geq 1}\) is a given sequence of integers and \(\alpha\) is a fixed real, are studied. In particular, the behaviour of the discrepancy of these sequences is considered from a metric point of view.NEWLINENEWLINEIn the introduction, some well-known results concerning upper and lower bounds of the discrepancy are presented. The main results of this paper show that for a large class of sequences \((a_n)_{n \geq 1}\) the discrepancy of the sequence \((\{a_n \alpha\})_{n \geq 1}\) has roughly an asymptotic order of \(\displaystyle {1 \over \sqrt{N}}\).NEWLINENEWLINEIn Theorem 1, \(P \in \mathbb{Z}[x]\) is a polynomial of degree \(d \geq 2\) and \((m_n)_{n\geq 1}\) is an arbitrary sequence of pairwise different integers with \(| m_n | \leq n^t\) for some \(t \in \mathbb{N}\) and for all \(n \geq n(t)\). Then, the discrepancy \(D_n\) of the sequence \((P(m_n) \alpha)_{n \geq 1}\) for almost \(\alpha\) satisfies the lower bound \(N D_N \geq N^{{1 \over 2} - \varepsilon}\) for all \(\varepsilon\) and for infinitely many values of \(N\).NEWLINENEWLINETheorem 1 is a consequence of the statement of Theorem 2. In Theorem 2, the function \(f: \mathbb{N} \to \mathbb{Z}\) satisfies \(| f(n) | \leq n^t\) for some \(t \in \mathbb{N}\) and all \(n \geq n(t)\). It is assumed that the set \(A_f(n) = \{(x,y) \in \mathbb{N}^2: f(x) + f(y) = n\}\) for all \(\varepsilon>0\) satisfies \(A_f(n) = {\mathcal O}(| n |^\varepsilon)\) as \(| n | \to \infty\). Then, for almost all \(\alpha\) the discrepancy of the sequence \((\{f(n) \alpha\}){n \geq 1}\) satisfies the lower bound \(N D_N \geq N^{{1 \over 2} - \varepsilon}\) for all \(\varepsilon\) and for infinitely many values of \(N\).NEWLINENEWLINEIn Theorem 3, \((a_n)_{n \geq 1}\) is a sequence of integers such that for some \(t \in \mathbb{N}\) the inequality \(| a_n | \leq n^t\) holds for all \(n\) large enough. It is assumed the there exists a number \(\tau \in (0,1)\) and strictly increasing sequence \((B_L)_{L \geq 1}\) of positive integers with \((B^\prime)^L \leq B_L \leq B^L\) for some reals \(B^\prime\) and \(B\) with \(1<B^\prime < B\) such that for all \(\varepsilon> 0\) and all \(L \geq L(\varepsilon)\) the inequality NEWLINE\[NEWLINE \int_0^1 \left| \sum_{n=1}^{B_L} e^{2 \pi {\mathbf i} a_n \alpha} \right| d \alpha \geq B_L^{\tau - \varepsilon} NEWLINE\]NEWLINE holds. Then, for almost all \(\alpha \in [0,1)\) and for all \(\varepsilon> 0\) the discrepancy \(D_N\) of the sequence \((\{a_n \alpha\}){n \geq 1}\) satisfies the low bound \(N D_N \geq N^{\tau - \varepsilon}\) for infinitely many values of \(N\).NEWLINENEWLINETheorems 4 and 5 are results from metric Diophantine approximation. In Theorem 4, \((R_L)_{L \geq 0}\) denotes a sequence of measurable subsets of \([0,1)\) with an one-dimensional Lebesgue measure \(\displaystyle \mathbb{P}(R_L) \geq {1 \over B^L}\) for some constant \(B \in \mathbb{R}^+\). Then, for almost all \(\alpha \in [0,1)\) for every \(\eta>0\) there are infinitely many integers \(h_L\) with \(\displaystyle h_L \leq (1 + \eta)^L{1 \over \mathbb{P}(R_L) }\) and \(\{h_L \alpha\} \in R_l\).NEWLINENEWLINETheorem 5 states that for every integer \(d \geq 1\) there is a strictly increasing sequence \((a_n)_{n \geq 1}\) of integers with \( {\displaystyle a_{n+1} \over a_n} \geq 1 + {c \over a_n^{1 \over d}}\) for some \(c > 0\) and all \(n \in \mathbb{N}\) such that for almost all \(\alpha\) the discrepancy \(D_N\) of the sequence \((\{a_n \alpha\}){n \geq 1}\) satisfies \(N D_N = {\mathcal O}((\log N)^{2 + \varepsilon})\) for all \(\varepsilon>0\).