The goal of this note is to provide a general lower bound on the number even values Fourier coefficients an arbitrary eta-quotient [Formula: see text], over any arithmetic progression. Namely, if text] denotes in degrees (mod text]) such that then we show unbounded for large. Note our result very close best currently known special case partition function (namely, proven by Bellaïche and Nicolas...

We present a natural restriction of Hindman’s Finite Sums Theorem that admits a simple combinatorial proof (one that does not also prove the full Finite Sums Theorem) and low computability-theoretic and proof-theoretic upper bounds, yet implies the existence of the Turing Jump, thus realizing the only known lower bound for the full Finite Sums Theorem. This is the first example of this kind. In...

Consider the finite regular language Ln = {w0 | w ∈ {0, 1}, |w| ≤ n}. In [3] it was shown that while this language is accepted by a deterministic finite automaton of size O(n), any one-way quantum finite automaton (QFA) for it has size 2 . This was based on the fact that the evolution of a QFA is required to be reversible. When arbitrary intermediate measurements are allowed, this intuition bre...

The problem of calculating exact lower bounds for the number k-faces d-polytopes with n vertices, each value k, and characterizing minimizers has recently been solved not exceeding 2d. We establish corresponding result $n=2d+1$; nature minimizing polytopes are quite different in this case. As a byproduct, we also characterize all $d+3$ vertices only one or two edges more than minimum.

We study the composition question for bounded-error randomized query complexity: Is R(f ◦ g) = Ω(R(f)R(g)) for all Boolean functions f and g? We show that inserting a simple Boolean function h, whose query complexity is only Θ(logR(g)), in between f and g allows us to prove R(f ◦ h ◦ g) = Ω(R(f)R(h)R(g)). We prove this using a new lower bound measure for randomized query complexity we call rand...

This study presents a novel technique to estimate the computational complexity of sequential decoding using the Berry-Esseen theorem. Unlike the theoretical bounds determined by the conventional central limit theorem argument, which often holds only for sufficiently large codeword length, the new bound obtained from the Berry-Esseen theorem is valid for any blocklength. The accuracy of the new ...