A variation of the prime k-tuples conjecture with applications to quantum limits

Let $$\mathcal {H}^{*}=\{h_1,h_2,\ldots \}$$ H ∗ = { h 1 , h 2 , … } be an ordered set of integers. We give sufficient conditions for the existence of increasing sequences of natural numbers $$a_j$$ a j and $$n_k$$ n k such that $$n_k+h_{a_j}$$ n k + h a j is a sum of two squares for every $$k\ge 1$$ k ≥ 1 and $$1\le j\le k.$$ 1 ≤ j ≤ k . Our method uses a novel modification of the Maynard–Tao sieve together with a second moment estimate. As a special case of our result, we deduce a conjecture due to D. Jakobson which has several implications for quantum limits on flat tori.

A new factor theorem on absolute matrix summability method

In this paper, we have a new matrix generalization with absolute matrix summability factor of an infinite series by using quasi-β-power increasing sequences. That theorem also includes some new and known results dealing with some basic summability methods

Some Problems on Approximations of Functions (Signals) in Matrix Summability of Legendre Series

In this paper, we prove a main theorem dealing the matrix summability of Legendre series using non-negative monotonic non-increasing sequences of function. This paper is more general than [9], [12] and [22].

Convergence of equilibrium measures corresponding to finite subgraphs of infinite graphs: New examples

A problem from thermodynamic formalism for countable symbolic Markov chains is considered. It concerns asymptotic behavior of the equilibrium measures corresponding to increasing sequences of finite submatrices of an infinite nonnegative matrix A A when these sequences converge to A A . After reviewing the results obtained up to now, a solution of the problem is given for a new matrix class. The geometric language of loaded graphs is used, instead of the matrix language.