Interplay between moments of difference of character values and alternatively signed binomial coefficients

TL;DR

This work investigates the interplay between moments of differences of character values and alternating binomial coefficients. It derives explicit asymptotic formulas for the $2m$-th power moments of differences such as $|\chi(f(n)) - \chi(f(n+1))|$ and $|e(f(n)) - e(f(n+1))|$, under non-degeneracy conditions on the polynomial $f$ and the modulus, using partial summation, Weyl bounds, and Weil-type estimates. A central result concerns the minimal subset $\mathcal{S}(N)$ needed to bound the alternating binomial sum by $(1+o(1))N^{1/2}$, proving that when $4|N$ the ratio $\mathcal{S}(N)/(N-1)\to 1$, i.e., almost all residues contribute. The paper provides detailed main-term constants and power-saving error terms across multiplicative, additive, and additive-character settings, connecting combinatorial questions about binomial sums to analytic number theory techniques and highlighting potential extensions to general finite fields.

Abstract

This article deals with finding the size of a set with smallest cardinality such that alternatively signed binomial coefficients sum up to $\sqrt{N}$. We deal with a character sum, where we capture the asymptotic formula for the moments of difference of characters over consecutive values on some dyadic interval of some suitable size $N$. Such sum plays a crucial role in the study of sum of binomial coefficients.

Theorems & Definitions (15)

• Theorem 1.1
• Theorem 2.1
• Theorem 2.2
• Theorem 2.3
• Theorem 3.1
• Lemma 3.2
• proof
• Lemma 3.3
• proof
• Theorem 4.1