Dear all,
The talk this week is by David Bradley of the University of Maine. Please note the special time. Since Professor Bradley is located in the US, we are starting later than usual.
This will be the final talk of the year. We will come back next year with a Ramanujan special, and hope that we get an opportunity to meet in person in the upcoming conference and holiday season. We wish you happy holidays and a great new year.
The announcement is as follows.
Talk Announcement:
Title: On Fractal Subsets of Pascal's "Pyramid" and the Number of Multinomial Coefficients Congruent to a Given Residue Modulo a Prime
Speaker: David Bradley (University of Maine, USA)
When: Thursday, Thursday Dec 7 23, 2023 - 6:30 PM (IST) (8AM EST/ 2PM (CET))
Where: Zoom: Ask the organisers for a link
Abstract.
We obtain an explicit formula and an asymptotic formula for the number of multinomial coefficients which are congruent to a given residue modulo a prime, and which arise in the expansion of a multinomial raised to any power less than a given power of that prime. Each such multinomial coefficient can be associated with a certain Cartesian product of intervals contained in the unit cube. For a fixed prime, the union of these products forms a set which depends on both the residue and the power of the prime. In the limit as the power of the prime increases to infinity, the sequence of unions converges in the Hausdorff metric to a non-empty compact set which is independent of the residue. We calculate the fractal dimension of this limiting set, and consider its monotonicity properties as a function of the prime. To study the relative frequency of various residue classes in the sequence of approximating sets, it would be desirable to have a closed-form formula for the number of entries in the first p rows of Pascal’s "pyramid" which are congruent to a given nonzero residue r modulo the prime p. Unfortunately, numerical computations with large prime moduli suggests that if there is such a formula, it is extremely complicated. Nevertheless, the evidence indicates that for sufficiently large primes p, the number of binomial coefficients in Pascal's triangle which are congruent to r mod p for r = 1, 1 < r < p−1, and r = p−1 is well approximated by the respective linear functions of p given
by 3p, p/2, and p. In particular, for large primes p there are approximately six times as many occurrences of the residue 1 in the first p rows of Pascal’s triangle reduced modulo p than there are of any other residue r in the range 1 < r < p − 1, and three times as many as r = p − 1. On the other hand, if we let the nonnegative integer k vary while keeping the prime p fixed, and look at the relative frequency of various residue classes that occur in the first pk rows, the seemingly substantial differences in frequency between r = 1, 1 < r < p−1, and r = p−1 when k = 1 are increasingly dissipated as k grows without bound. We show that in the limit as k tends to infinity, all nonzero residues are equally represented with asymptotic proportion 1/(p − 1).
