Kaimin Cheng

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20242026

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9 papers · 1 filter

math.NT2026

Counting subspaces with prescribed Welch sums

Kaimin Cheng, Jie Hu

Let $q = 2^{n}$ with $n = 2 m + 1$ for an integer $m \geq 1$ , and let $F_{q}$ be the finite field with $q$ elements. Let $W_{n} (x) = x^{2^{m} + 3}$ be the Welch function on $F_{q}$ . Fo…

math.NT2026

Counting zero-sum subspaces for the multiplicative inverse function

Kaimin Cheng

Let $q = 2^{n}$ with $n \geq 6$ , and let $F_{q}$ be the finite field with $q$ elements. For a subspace $E$ of $F_{q}$ , define \[ S(E)=\sum_{x\in E\setminus\{0\}}x^{-1}.\] Le…

math.NT2026

Sharp extremal asymptotics for Cusick's sum-of-digits bias at fixed Hamming weight

Kaimin Cheng

Let $s_{2} (n)$ be the binary sum-of-digits function and let $c_{t}$ be the natural density of the integers $n \geq 0$ for which $s_{2} (n + t) \geq s_{2} (n)$ . Earlier work of the author proved the…

math.CO2026

Deficit bounds and equality cases in the Tu--Deng problem

Kaimin Cheng

Let $N = 2^{k} - 1$ , and let $S_{t, k}$ consist of the pairs $0 \leq a, b < N$ such that $a + b \equiv t (mod N)$ and $\wt(a)+\wt(b)<k$. The Tu--Deng bound $∣ S_{t, k} ∣ \leq 2^{k - 1}$ has recently been…

math.NT2026

The complete cubic Walsh spectrum of a permutation-inverse Boolean family

Kaimin Cheng

Let $q = 2^{e}$ with $e \geq 2$ even, put $d = (q^{2} + q + 1) /3$ , and let $σ (X) = X + X^{d} + X^{d q}$ be the permutation of $F_{q^{2}}$ introduced by Ding, Qu, Wang, Yuan, and Yuan. For $α\in\math…

math.NT2026

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…

math.NT2026

Banded quadratic digit functions along irreducible polynomials over finite fields

Kaimin Cheng

Let $q$ be an odd prime power and let $\F_q$ be the finite field with $q$ elements. Let $P (n)$ be the set of monic irreducible polynomials of degree $n$ over $\mathbb{F}_…

math.NT2026

On the $2$ -adic valuation of $σ_{k} (n)$

Kaimin Cheng, Ke Zhang

For a positive integer $k$ , let \[ σ_k(n)=\sum_{d\mid n} d^k \] be the divisor function of order $k$ , and let $ν_{p} (m)$ denote the $p$ -adic valuation of an integer $m$ . Motivated by…

math.NT2026

A proof of a permutation-inverse bent-function conjecture

Kaimin Cheng

Let $q = 2^{e}$ with $e$ even, and let $F_{q^{2}}$ be the finite field of order $q^{2}$ . Put $d = (q^{2} + q + 1) /3$ , and consider the permutation polynomial $$ σ(X)=X+X^d+X^{dq}\in\mathbb…

Counting subspaces with prescribed Welch sums

Counting zero-sum subspaces for the multiplicative inverse function

Sharp extremal asymptotics for Cusick's sum-of-digits bias at fixed Hamming weight

Deficit bounds and equality cases in the Tu--Deng problem

The complete cubic Walsh spectrum of a permutation-inverse Boolean family

A first-exit proof of Cusick's sum-of-digits conjecture

Banded quadratic digit functions along irreducible polynomials over finite fields

On the 2-adic valuation of σk​(n)

A proof of a permutation-inverse bent-function conjecture

On the $2$ -adic valuation of $σ_{k} (n)$