We revisit a classical result of Coppersmith and Shparlinski showing bounds on how well polynomials can approximate the discrete log function. They showed that for any univariate polynomial $f(X) \in \mathbb{F}_p[X]$ with $deg(f) \leq D$, we have: $$ |\{ x \in \mathbb{F}_p^\times \mid f(x) = dlog(x) \}| \leq \sqrt{2 D p}.$$ We show two improvements: (1) For all $D$, we improve the upper bound above to $(1+o(1)) \cdot \sqrt{Dp}$. In particular, we give nontrivial upper bounds on the approximation even for $D>p/2$. This new upper bound is the Johnson bound from the theory of list-decoding of error-correcting codes, and in fact our proof uses a connection to that problem, as well as other combinatorial ideas. (2) For $D = 1$, where both the above bounds are $O(\sqrt{p})$, we give an asymptotically improved upper bound of $O(p^{2/5})$. This is based on the sum-product phenomenon of additive combinatorics. We also improve the [CS00] bounds on the number of points of the form $(x, dlog(x)) \in \mathbb{F}_p^2$ that any bivariate polynomial $Q(X,Y) \in \mathbb{F}_p[X,Y]$ with $deg(Q) \leq D$ can vanish on, with our new bounds approaching the analogue of the Johnson bound for this problem.
We present an exposition of an argument, discovered by GPT-6 Pro, that gives an optimal bound for the polynomial Littlewood-Offord problem. Namely, let $F$ be a degree-$d$ multilinear polynomial that contains $r$ degree-$d$ monomials involving disjoint sets of variables. Then, for i.i.d. Rademacher random variables $\x...
We study the problem of finding tight explicit bounds on sums over zeros of $L$-functions near the $1$-line with a variety of zero-free regions and zero-density estimates. As an application to demonstrate the techniques, we apply them to the case of the Riemann zeta function and the prime number theorem to obtain new b...
Let $q=p^f$, where $p$ is prime. We prove that every $q$-ary linear code meeting the Griesmer bound, with positive dimension and minimum distance $d\ge1$, is $p^{\max\{0,\nu_p(d)-f+1\}}$-divisible, as asserted in Ward's conjecture. Arbitrary projective column multiplicities are allowed. The geometric ingredient is a bo...
For each real $b$, exact upper and lower bounds on the probability $\mathsf P(X\ge b)$ over all random variables $X$ with log-concave p.d.f.'s such that $\mathsf E X=0$ and $\mathsf E X^2=1$ are obtained, as well as the best constant factor $C$ in the inequality $\mathsf P(X\ge b)\le C e^{-b}$ for all real $b\ge0$. Exa...
We prove a folklore conjecture on exponential sums with multiplicative coefficients: if $f$ is multiplicative with $|f(n)|\le1$, $(a,q)=1$, $|\alpha-a/q|\le q^{-2}$ and $q\le N/(\log N)^2$, then \[ \sum_{n\le N}f(n)\operatorname{e}(n\alpha)\ll\frac N{\log N}+\frac N{\sqrt q}. \] Complete multiplicativity is not require...
Assume that the Riemann zeta function has no zeros in $\Re(s)>1-\delta$, $s\neq1$, for some fixed $0<\delta\leq1/2$. We prove that, for every sufficiently large integer $k$, a positive multiple $m_k k!$ has arithmetic complexity $\tau(m_k k!)\leq\exp(C\sqrt{\log k\log\log k})$, where $C$ depends only on $\delta$. We fi...
Liang Zhang· 0 citations
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