11 papers · 1 filter
Sharp extremal asymptotics for Cusick's sum-of-digits bias at fixed Hamming weight
Kaimin Cheng
Let be the binary sum-of-digits function and let be the natural density of the integers for which . Earlier work of the author proved the…
The complete cubic Walsh spectrum of a permutation-inverse Boolean family
Kaimin Cheng
Let with even, put , and let be the permutation of introduced by Ding, Qu, Wang, Yuan, and Yuan. For $α\in\math…
A first-exit proof of Cusick's sum-of-digits conjecture
Kaimin Cheng
We prove Cusick's conjecture on the binary sum-of-digits function. More precisely, for every integer \(t\ge 1\) we show that \[ c_t:=\lim_{N\to\infty}\frac{1}{N} \#\{0\le n<N:\ s_2…
Banded quadratic digit functions along irreducible polynomials over finite fields
Kaimin Cheng
Let be an odd prime power and let $\F_q$ be the finite field with elements. Let be the set of monic irreducible polynomials of degree over $\mathbb{F}_…
On the -adic valuation of
Kaimin Cheng, Ke Zhang
For a positive integer , let \[ σ_k(n)=\sum_{d\mid n} d^k \] be the divisor function of order , and let denote the -adic valuation of an integer . Motivated by…
A proof of a permutation-inverse bent-function conjecture
Kaimin Cheng
Let with even, and let be the finite field of order . Put , and consider the permutation polynomial $$ σ(X)=X+X^d+X^{dq}\in\mathbb…