Searcharxiv⌕ Search

arXiv subjects

Lukas Spiegelhofer

Publications and source records attributed to Lukas Spiegelhofer.

32 records · Page 2Linked to original sources
arXiv2018

The level of distribution of the Thue--Morse sequence

The level of distribution of a complex valued sequence $b$ measures "how well $b$ behaves" on arithmetic progressions $nd+a$. Determining whether $θ$ is a level of distribution for $b$ involves summing a certain error over $d\leq D$, where $D$ depends on $θ$, this error is given by comparing a finite sum of $b$ along $nd+a$ and the expected value of the sum. We prove that the Thue--Morse sequence has level of distribution $1$, which is essentially best possible. More precisely, this sequence gives one of the first nontrivial examples of a sequence satisfying a Bombieri--Vinogradov type theorem for each exponent $θ<1$. In particular, this result improves on the level of distribution $2/3$ obtained by Müllner and the author. As an application of our method, we show that the subsequence of the Thue--Morse sequence indexed by $\lfloor n^c\rfloor$, where $1<c<2$, is simply normal. That is, each of the two symbols appears with asymptotic frequency $1/2$ in this subsequence. This result improves on the range $1<c<3/2$ obtained by Müllner and the author and closes the gap that appeared when Mauduit and Rivat proved (in particular) that the Thue--Morse sequence along the squares is simply normal. In the proofs, we reduce both problems to an estimate of a certain Gowers uniformity norm of the Thue--Morse sequence similar to that given by Konieczny (2017).

math.NT↗
arXiv2017

A digit reversal property for Stern polynomials

We consider the following polynomial generalization of Stern's diatomic series: let $s_1(x,y)=1$, and for $n\geq 1$ set $s_{2n}(x,y)=s_n(x,y)$ and $s_{2n+1}(x,y)=x\,s_n(x,y)+y\,s_{n+1}(x,y)$. The coefficient $[x^iy^j]s_n(x,y)$ is the number of hyperbinary expansions of $n-1$ with exactly $i$ occurrences of the digit $\mathtt 2$ and $j$ occurrences of $\mathtt 0$. We prove that the polynomials $s_n$ are invariant under \emph{digit reversal}, that is, $s_n=s_{n^R}$, where $n^R$ is obtained from $n$ by reversing the binary expansion of $n$.

math.NT↗
arXiv2017

Normality of the Thue--Morse sequence along Piatetski-Shapiro sequences, II

We prove that the Thue--Morse sequence $\mathbf t$ along subsequences indexed by $\lfloor n^c\rfloor$ is normal, where $1<c<3/2$. That is, for $c$ in this range and for each $ω\in\{0,1\}^L$, where $L\geq 1$, the set of occurrences of $ω$ as a subword (contiguous finite subsequence) of the sequence $n\mapsto \mathbf t_{\lfloor n^c\rfloor}$ has asymptotic density $2^{-L}$. This is an improvement over a recent result by the second author, which handles the case $1<c<4/3$. In particular, this result shows that for $1<c<3/2$ the sequence $n\mapsto \mathbf t_{\lfloor n^c\rfloor}$ attains both of its values with asymptotic density $1/2$, which improves on the bound $c<1.4$ obtained by Mauduit and Rivat (who obtained this bound in the more general setting of $q$-multiplicative functions, however) and on the bound $c\leq 1.42$ obtained by the second author. In the course of proving the main theorem, we show that $2/3$ is an admissible level of distribution for the Thue--Morse sequence, that is, it satisfies a Bombieri--Vinogradov type theorem for each exponent $η<2/3$. This improves on a result by Fouvry and Mauduit, who obtained the exponent $0.5924$. Moreover, the underlying theorem implies that every finite word $ω\in\{0,1\}^L$ is contained as an arithmetic subsequence of $\mathbf t$.

math.NT↗
arXiv2017

An explicit generating function arising in counting binomial coefficients divisible by powers of primes

For a prime $p$ and nonnegative integers $j$ and $n$ let $\vartheta_p(j,n)$ be the number of entries in the $n$-th row of Pascal's triangle that are exactly divisible by $p^j$. Moreover, for a finite sequence $w=(w_{r-1}\cdots w_0)\neq (0,\ldots,0)$ in $\{0,\ldots,p-1\}$ we denote by $\lvert n\rvert_w$ the number of times that $w$ appears as a factor (contiguous subsequence) of the base-$p$ expansion $n=(n_{μ-1}\cdots n_0)_p$ of $n$. It follows from the work of Barat and Grabner (Digital functions and distribution of binomial coefficients, J. London Math. Soc. (2) 64(3), 2001), that $\vartheta_p(j,n)/\vartheta_p(0,n)$ is given by a polynomial $P_j$ in the variables $X_w$, where $w$ are certain finite words in $\{0,\ldots,p-1\}$, and each variable $X_w$ is set to $\lvert n\rvert_w$. This was later made explicit by Rowland (The number of nonzero binomial coefficients modulo $p^α$, J. Comb. Number Theory 3(1), 2011), independently from Barat and Grabner's work, and Rowland described and implemented an algorithm computing these polynomials $P_j$. In this paper, we express the coefficients of $P_j$ using generating functions, and we prove that these generating functions can be determined explicitly by means of a recurrence relation. Moreover, we prove that $P_j$ is uniquely determined, and we note that the proof of our main theorem also provides a new proof of its existence. Besides providing insight into the structure of the polynomials $P_j$, our results allow us to compute them in a very efficient way.

math.NT↗
arXiv2017

Divisibility of binomial coefficients by powers of two

For nonnegative integers $j$ and $n$ let $Θ(j,n)$ be the number of entries in the $n$-th row of Pascal's triangle that are not divisible by $2^{j+1}$. In this paper we prove that the family $j\mapstoΘ(j,n)$ usually follows a normal distribution. The method used for proving this theorem involves the computation of first and second moments of $Θ(j,n)$, and uses asymptotic analysis of multivariate generating functions by complex analytic methods, building on earlier work by Drmota (1994) and Drmota, Kauers and Spiegelhofer (2016).

math.NT↗
arXiv2017

Discrepancy results for the Van der Corput sequence

Let $d_N=ND_N(ω)$ be the discrepancy of the Van der Corput sequence in base $2$. We improve on the known bounds for the number of indices $N$ such that $d_N\leq \log N/100$. Moreover, we show that the summatory function of $d_N$ satisfies an exact formula involving a $1$-periodic, continuous function. Finally, we show that $d_N$ is invariant under digit reversal in base $2$.

math.NT↗
arXiv2017

A digit reversal property for an analogue of Stern's sequence

We consider a variant of Stern's diatomic sequence, studied recently by Northshield. We prove that this sequence $b$ is invariant under \emph{digit reversal} in base $3$, that is, $b_n=b_{n^R}$, where $n^R$ is obtained by reversing the base-$3$ expansion of $n$.

math.NT↗
arXiv2017

Piatetski-Shapiro sequences via Beatty sequences

Integer sequences of the form $\lfloor n^c\rfloor$, where $1<c<2$, can be locally approximated by sequences of the form $\lfloor nα+β\rfloor$ in a very good way. Following this approach, we are led to an estimate of the difference \[\sum_{n\leq x}φ\left(\lfloor n^c\rfloor\right)-\frac 1c\sum_{n\leq x^c}φ(n)n^{\frac 1c-1},\] which measures the deviation of the mean value of $φ$ on the subsequence $\lfloor n^c\rfloor$ from the expected value, by an expression involving exponential sums. As an application we prove that for $1<c\leq 1.42$ the subsequence of the Thue-Morse sequence indexed by $\lfloor n^c\rfloor$ attains both of its values with asymptotic density $1/2$.

math.NT↗
arXiv2017

Normality of the Thue--Morse sequence along Piatetski-Shapiro sequences

We prove that for $1<c<4/3$ the subsequence of the Thue--Morse sequence $\mathbf t$ indexed by $\lfloor n^c\rfloor$ defines a normal sequence, that is, each finite sequence $(\varepsilon_0,\ldots,\varepsilon_{T-1})\in \{0,1\}^T$ occurs as a contiguous subsequence of the sequence $n\mapsto \mathbf t\left(\lfloor n^c\rfloor\right)$ with asymptotic frequency $2^{-T}$.

math.NT↗
arXiv2017

Möbius orthogonality for the Zeckendorf sum-of-digits function

We show that the (morphic) sequence $(-1)^{s_φ(n)}$ is asymptotically orthogonal to all bounded multiplicative functions, where $s_φ$ denotes the Zeckendorf sum-of-digits function. In particular we have $\sum_{n<N} (-1)^{s_φ(n)} μ(n) = o(N)$, that is, this sequence satisfies the Sarnak conjecture.

math.NT↗
arXiv2016

Pseudorandomness of the Ostrowski sum-of-digits function

For an irrational $α\in(0,1)$, we investigate the Ostrowski sum-of-digits function $σ_α$. For $α$ having bounded partial quotients and $\vartheta\in\mathbb R\setminus\mathbb Z$, we prove that the function $g:n\mapsto \mathrm e(\vartheta σ_α(n))$, where $\mathrm e(x)=\mathrm e^{2πi x}$, is pseudorandom in the following sense: for all $r\in\mathbb N$ the limit \[γ_r= \lim_{N\rightarrow\infty}\frac 1N\sum_{0\leq n<N}g(n+r)\overline{g(n)} \] exists and we have \[\lim_{R\rightarrow\infty}\frac 1R\sum_{0\leq r<R}\bigl\lvert γ_r\bigr\rvert^2=0.\]

math.NT↗
arXiv2016

On a Conjecture of Cusick Concerning the Sum of Digits of n and n + t

For a nonnegative integer $t$, let $c_t$ be the asymptotic density of natural numbers $n$ for which $s(n + t) \geq s(n)$, where $s(n)$ denotes the sum of digits of $n$ in base $2$. We prove that $c_t > 1/2$ for $t$ in a set of asymptotic density $1$, thus giving a partial solution to a conjecture of T. W. Cusick stating that $c_t > 1/2$ for all t. Interestingly, this problem has several equivalent formulations, for example that the polynomial $X(X + 1)\cdots(X + t - 1)$ has less than $2^t$ zeros modulo $2^{t+1}$. The proof of the main result is based on Chebyshev's inequality and the asymptotic analysis of a trivariate rational function, using methods from analytic combinatorics.

math.CO↗
arXiv2015

The maximal order of hyper-($b$-ary)-expansions

Using methods developed by Coons and Tyler, we give a new proof of a recent result of Defant, by determining the maximal order of the number of hyper-($b$-ary)-expansions of a nonnegative integer $n$ for general integral bases $b\geqslant 2$.

math.CO↗