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Construction of Šindel sequences

Michal Křížek, Alena Šolcová, Lawrence Somer (2007)

Commentationes Mathematicae Universitatis Carolinae

We found that there is a remarkable relationship between the triangular numbers T k and the astronomical clock (horologe) of Prague. We introduce Šindel sequences { a i } ⊂ ℕ of natural numbers as those periodic sequences with period p that satisfy the following condition: for any k ∈ ℕ there exists n ∈ ℕ such that T k = a 1 + ⋯ + a n . We shall see that this condition guarantees a functioning of the bellworks, which is controlled by the horologe. We give a necessary and sufficient condition for a periodic sequence to be a Šindel sequence....

Convergence of series of dilated functions and spectral norms of GCD matrices

Christoph Aistleitner, István Berkes, Kristian Seip, Michel Weber (2015)

Acta Arithmetica

We establish a connection between the L² norm of sums of dilated functions whose jth Fourier coefficients are ( j - α ) for some α ∈ (1/2,1), and the spectral norms of certain greatest common divisor (GCD) matrices. Utilizing recent bounds for these spectral norms, we obtain sharp conditions for the convergence in L² and for the almost everywhere convergence of series of dilated functions.

Coprimality of integers in Piatetski-Shapiro sequences

Watcharapon Pimsert, Teerapat Srichan, Pinthira Tangsupphathawat (2023)

Czechoslovak Mathematical Journal

We use the estimation of the number of integers n such that ⌊ n c ⌋ belongs to an arithmetic progression to study the coprimality of integers in ℕ c = { ⌊ n c ⌋ } n ∈ ℕ , c > 1 , c ∉ ℕ .

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